Precalculus 1: Basic notions

Explore the essential mathematical symbols and Greek letters you'll encounter, learn where they appear across arithmetic, equations, logic, and set theory, and master the notation behind limits and absolute values.

Discover how Greek letters appear in math across general usage, geometry, calculus, linear algebra, and probability—from pi, sigma, epsilon, delta to alpha, theta, lambda, mu, and omega.

Latin letters in mathematics illustrate common uses: a,b,c real numbers, triangle sides, e Euler’s number, f,g,h functions, and i imaginary unit or index, with x,y,z variables.

Trace the progression from natural numbers to complex numbers, explaining why we need integers, rationals, irrationals, and reals. Use Venn diagrams to show their relationships and arithmetic foundations.

Define natural numbers as an ordered, discrete set closed under addition and multiplication, with zero and one as identities, and learn terms, factors, product, factorization, and multiplication’s precedence over addition.

Explore the commutativity of addition and multiplication for natural numbers with practical illustrations, showing how order independence motivates these axioms for real numbers.

Explore the associativity of addition and multiplication for natural numbers, showing that grouping does not affect the sum or product of three numbers, with intuitive counting and volume illustrations.

Explore the distributive law, its geometric and arithmetic illustrations, and its consequences, including expanding (k+l)(m+n) into km+kn+lm+ln, with area and price examples.

Explore divisors and prime numbers, define divisors, and distinguish primes from composites. Learn prime factorization and practical divisibility rules for 2, 3, and 5.

Explore prime factorization by factoring numbers such as 60, 105, and 48, and see how it underpins adding fractions, least common denominators, and polynomial factorization.

Use the Sieve of Eratosthenes to find all prime numbers up to a given number by marking primes and deleting multiples up to the square root, illustrated with 121.

Review integer arithmetic by exploring addition, subtraction, and multiplication, the zero as the neutral element of addition, opposite numbers, and subtraction as adding the opposite on the real number axis.

Explore how rational numbers become fractions—numerator and denominator—and learn irreducible, proper, improper, and mixed forms, plus addition, subtraction, multiplication, division, and comparison using the least common denominator.

Explore decimal expansion of rational numbers and its role in distinguishing rational from irrational numbers, with practical steps for converting decimals to fractions and comparing values.

Explore finite and periodic decimal expansions and how they distinguish rational from irrational numbers. See examples like 1/2 = 0.5, 1/3 = 0.333..., and 0.999... = 1.

Demonstrates converting decimal expansions to irreducible fractions using finite and periodic forms, showing x=3/10, y=1/50, and z=1/3 via the ten times minus z method.

Represent a finite and periodic decimal with period two as a fraction by using 100x minus x, yielding 99x = 27 and x = 3/11.

Demonstrate converting finite and periodic decimal expansions to an irreducible fraction using a repeating-nines technique, solving 900x = 135 to get x = 0.15, and that periodic decimals are rational.

Discover irrational numbers as real numbers with infinite, nonperiodic decimal expansions, contrast them with rationals, and explore their non-closure under addition, subtraction, multiplication, and division.

Explore the distributive law and derive the square of the sum and the square of the difference, showing how these formulas underpin completing the square in precalculus.

Show that with a = sqrt(2) and b = sqrt(8), the square of the difference equals 2, demonstrating a rational number via square-root rules under positivity.

Apply the distributive law to (a - b)(a + b) to derive a^2 - b^2, showing ab terms cancel; memorize difference of two squares formula for solving quadratic equations.

Master the order of operations and pemdas, and learn to treat division as multiplication by inverse and subtraction as addition of opposites, with attention to signs and parentheses.

Discover that the distributive law and precedence rules don’t contradict each other; use the distributive law to simplify expressions and choose the optimal method, sometimes avoiding computing inside parentheses first.

Solve problem ten on fractions by expressing the result as an irreducible fraction and applying two methods with common denominators, noting the distributive law and avoiding splitting the denominator.

Apply the square of sum and square of difference formulas to simplify a nonzero A and B expression, revealing a constant 4/5 regardless of A and B.

From factoring and canceling, enforce A ≠ 0, B ≠ 0, A ≠ B, and A ≠ -B, yielding the final result A^3 B^7 / (2A - B).

Determine the domain of a two-variable function by excluding x=0, y=0, and x=y, then simplify to (x+y)/(3xy) and show a continuous extension on y=x but not on the axes.

Explore values of a function of two variables by plugging in (-1,-1), (1,-1), and (0,-1) to a given formula, practicing basic arithmetic for precalculus to calculus three.

Explore the binomial-like formula a^n − b^n = (a − b) sum_{k=0}^{n−1} a^{n−1−k} b^k, with proofs, small-n cases, and sigma notation for the derivative of polynomial functions in calculus.

Derive the derivative of a polynomial of degree n using the difference quotient, confirming that d/dx x^n = n x^{n-1} and illustrating with n=3.

Explore the concept of distance between real numbers and points on the plane, and see why proximity underpins limits, continuity, and derivatives in calculus.

Explore the distance between real numbers using absolute value, define the distance function d(x,y)=|x−y|, and review key properties like non-negativity, |x|=|−x|, and |xy|=|x||y|.

Explore the Cartesian coordinate system in the plane: locate points with the x and y axes and grid lines, and understand reflections about the axes and origin in R2.

Learn to compute the Euclidean distance between two points in the plane using the Pythagorean theorem, covering horizontal, vertical, and diagonal cases with the distance formula.

Discover the triangle inequality in real numbers and vectors, including the absolute value form |x+y| ≤ |x|+|y|, and its essential role in calculus through proofs of limit rules.

define metric spaces and distance functions, and prove the triangle inequality, with examples including discrete metrics, real-number distance via absolute value, and euclidean distance in the plane.

Explore the taxicab distance (Manhattan distance) between points in the plane, defined as the sum of absolute differences. Verify positivity, symmetry, and triangle inequality to confirm it is a metric.

Explore the max distance in the plane, d infinity, defined as the maximum of |x1 - x2| and |y1 - y2|, and link to Manhattan (D1) and Euclidean (D2) distances.

Define open balls and punctured neighborhoods in metric spaces, center-radius distance notation, and illustrate with real line and Euclidean plane open disks.

Explore how the unit circle in the Manhattan (taxicab) metric forms a diamond shape, defined by |x|+|y|=1, and contrast it with disks and open balls.

Explore the unit circle in the max metric, where distance equals the larger of the absolute differences of x and y coordinates; reveal a square boundary and its open ball.

Discover how distance underpins calculus by defining limits across one-variable, vector-valued, and multivariable functions, using Euclidean distance and metric spaces to connect continuity, derivatives and partial derivatives, and integrals.

Explore equations and inequalities, establishing terminology around unknowns, solutions or roots, and solution sets across linear, nonlinear, circle, and matrix equations.

Introduces inequalities and solution sets as intervals on the real line, teaching interval notation, graphing intervals, and how brackets, parentheses, and braces distinguish closed, open, and two-element sets.

Learn the difference between solving and verifying solutions, then see how for x^2+2x-3, roots are -3 and 1, and the inequality yields (-3,1).

Identify one-variable equations and distinguish linear equations from polynomial, trigonometric, exponential, and logarithmic types, then learn to reduce linear equations to ax = -b/a using coefficients.

Apply operations that preserve the solution set, such as adding or subtracting the same amount on both sides, or multiplying by a nonzero number, to solve for x.

Explore inconsistent equations and false roots through concrete examples, showing when no solution, all real numbers, and how domain restrictions create false roots.

Explore linear equations with fractions and absolute value. Apply distributive law, cancel terms, and solve for x, yielding x = 1/15.

solve linear equations with absolute value by locating turning points and splitting into cases, with turning points at 1 and 5; use distance interpretation and verify solutions.

Solve linear equations with absolute value by breaking the domain into intervals at -1 and 2, substitute to remove absolute values, and solve case-wise; obtain -2, 0, and 4.

Solve linear equations with absolute value by case analysis across intervals defined by breaking points -1 and 2, yielding solutions x = -4 and x = 2.

Solve linear equations with absolute value by interval casework; verify roots in each case, yielding x = -5 as the only solution.

Solve linear equations with absolute value via three-case analysis, define left-hand side as a function, and graph it to compare with right-hand side, yielding the solution set [-1, 2].

Explore future precalculus topics through concrete examples of equations and their solution sets, including linear, quadratic, and multi-variable cases with geometric interpretations.

Learn how to describe a line in the plane using slope and intercept, including horizontal and vertical cases, and how to find the slope and y-intercept from two points.

Derive the slope-intercept form for a line through two points using multiple methods, compute m and b, and verify with coordinates or by reading from a picture.

Learn which operations preserve a linear inequality solution set: add or subtract the same amount, and multiply or divide by a positive number; negative numbers reverse the inequality.

Explore how multiplying inequalities by variable expressions can reverse or preserve the sign, using a visualization, and learn to solve rational inequalities by combining fractions and analyzing signs.

Solve a linear inequality by applying the distributive law and combining like terms, then isolate x to obtain the solution set [5/4, infinity], illustrating the interval form.

Explore how to solve inequalities with absolute value using distance from zero, geometric interpretation, and interval notation, including cases with multiple absolute values and general f(x).

Solve the linear inequality with absolute value |3-2x| ≤ 5 using algebraic and geometric methods, obtaining x in [-1, 4].

solve linear inequalities with absolute value by splitting into two cases, then describe the solution set as x<-2 or x>5, shown via distance reasoning on a number line.

Identify turning points and split the axis into cases to solve linear inequalities with absolute values by replacing expressions with their corresponding cases in each interval.

Learn to solve equations and inequalities in one variable to locate zeros and sign regions, and use derivatives and second derivatives to identify maxima, minima, and convexity in precalculus.

Motivate studying equations and inequalities in calculus three, showing how two-variable descriptions yield curves and regions, interpret arithmetical descriptions as domains, and apply to double integrals and optimization.

Define the unit disk in taxicab distance by |x|+|y| ≤ 1, and learn to draw its boundary with lines of slope 1 or -1 in each quadrant.

Explore solving systems of linear equations using graphical, substitution, and elimination methods, and recognize cases with unique, infinite, or no solutions, essential for partial fraction decomposition in calculus two.

Explore function as a relation between independent and dependent variables, with x or t as argument and y as value, including a linear example d(t)=6t and domain.

Explore the function as a relation between independent and dependent variables, illustrate with f(x)=x^2 and its parabola, graph functions, and apply the vertical-line test.

Explore functions as relations between variables, comparing explicit definitions with implicit ones, and apply the vertical line test to circles and lemniscates, revealing symmetries and implicit graphing challenges.

Explore functions you’ll study in precalculus and calculus, from polynomial, linear, and trigonometric to exponential and logarithmic forms, plus multi-variable graphs and vector fields.

Learn how to think about sets and describe them verbally, graphically with Venn diagrams, or by roster and set-builder notation, and analyze unions, intersections, and differences.

Explore how domain, codomain, and range define a function, with each input x mapping to a unique output y, and visualize f as a black box.

Explore describing a function by formula and identifying its domain, codomain, and range, using a linear example on (-1, 1], and compute its x-intercept and y-intercept.

Identify three domain issues: avoid division by zero, even-degree roots of negative numbers, and logarithms of nonpositive values. Use 1/x and sine x to illustrate domain and range.

Analyze functions from pictures to determine domain, range, and x- and y-intercepts for functions f and g, illustrating two graph-based descriptions.

Describe a function using a table (way three) between finite sets, showing domain, codomain, and range with an example mapping a, b, e, f to 1, 1, 4, 2.

Explore four ways to describe functions, focusing on the verbal description via two real-life examples: Sweden's social security numbers and assigning birth mothers in a village.

Explore surjections, injections, and bijections, and learn how to detect injections using formulas, the horizontal line test, and tables or graphs. Illustrate when functions are onto, one-to-one, or both.

Explore inverse functions and how a bijection enables reversing a function, with notation f inverse and graph symmetry about the line y=x, plus examples like linear and reciprocal functions.

Explore monotone functions by defining increasing, decreasing, non-decreasing, and non-increasing types, illustrate with floor and ceiling functions, and discuss interval monotonicity and derivative notes.

Explore bounded functions, defined by values staying within a fixed range, and distinguish bounded above and below, maximum and minimum—local and global—plus infimum and supremum, with sine x and x^2.

Explore adding, scaling, subtracting, multiplying, and dividing two functions on the same domain to form new functions. See examples with f(x)=x+5 and g(x)=x^2−1, including linear combinations and rational functions.

Discover the composition of functions, define inner and outer functions, and understand g circle f notation, order, and domain, with examples and relevance to the chain rule.

Explore why the order of functions in a composition matters, applying the chain rule to derivatives of composed functions and distinguishing inner and outer functions.

Learn about even and odd functions, their y-axis and origin symmetry, and how cosine and sine illustrate these properties. Use integration by inspection to evaluate symmetric integrals quickly.

Explore transformations of graphs by shifting, reflecting, scaling parabolas, and applying absolute value, plus an intro to completing the square for precalculus two.

Explore various operations on linear functions f and g to produce new functions by addition, linear combinations, products, and both compositions f∘g and g∘f.

Determine the domain of a rational function and compute its inverse, then verify that f inverse composed with f yields the identity, and inspect intercepts.

Compare two similar functions where g is y = x+1 with full domain, while f equals x+1 but excludes x=1, removing the point (1,2); illustrate domain and range differences.

Explore a piecewise function defined on three intervals, determine its domain and range, plot its graph, and assess properties like monotonicity, boundedness, evenness, and invertibility.

Piecewise function with left y=x on [-2,-1), middle y=x^2 on [-1,1], right y=-x+4 on (1,2); range [-2,-1) ∪ [0,1] ∪ (2,3); discontinuous at -1 and 1; not injective.

Examine a piecewise function featuring a parabola and constant segments, where the domain is a union of intervals with endpoints excluded and the range and invertibility are analyzed.

Explore how absolute value creates piecewise functions and apply a functional approach to inequalities. Plot f(x)=|x| and compare with g1–g4 for problem 7 to read intersections.

Explore graph transformations, including vertical and horizontal shifts, reflections, and scaling, while noting that adding to the argument causes a left shift and detailing domain–range changes.

Transform graph operations on f: push down, shift right, scale by one half, squash horizontally, reflect across the x-axis and y-axis, and apply absolute value to obtain y = |f(x)|.

Investigate the Heaviside step function, a piecewise on-off function, and apply graph transformations such as shifting by two and three steps, subtracting, and multiplying by the linear function 3 - x.

Classify functions as even, odd, or neither using f(-x)=f(x) for even and f(-x)=-f(x) for odd. Illustrate with x^4 (even), x^3 (odd), and examples that are neither.

Determine the domain and range of the composed function f(x) = sin(sqrt(x+1)); establish x ≥ -1 and a range from -1 to 1.

Determine the domain of f(x) = 1 / log(sin x) between even and odd multiples of pi, excluding zeros. The range is minus infinity to zero.

Explore a highly composed function built from identity, a polynomial, and power and root operations, ending with a minus six power; the video maps the construction for differentiation.

Differentiates a complex function by applying the derivative rules for sums, constants, chains, and powers, tracing from the outer to the inner operation using the function's structure.

demonstrate that for an odd function f and an even function g, f(0)=0 and the product fg is odd, while the sum f+g need not be odd.

Explore the definition of a statement as a declarative sentence with a true or false logical value, and distinguish open from closed statements in symbolic logic.

Explore the delta neighborhood of a, defined by |x - a| < δ, and distinguish open from punctured neighborhoods. Learn how negation reveals distance relations and their role in calculus.

Explore unary logical connectives with negation (not p), defined via truth tables, and practice negating statements like prime, equality, and ordering.

Learn how the conjunction and forms a compound statement and is true only when both parts are true, using truth tables and examples.

Study the disjunction or, the binary logical connective defined as p or q, which is true if at least one statement is true and false only when both are false.

Explore negation, conjunction, and disjunction with precedence rules and parentheses. The lecture walks through seven short problems, explaining truth values and evaluating complex statements.

Explore the binary logical connective implication, its truth table, and the concept of a necessary condition, with examples like divisibility and geometric and algebraic cases.

Define implication with its antecedent and consequent, then introduce the converse, inverse, and contrapositive. Demonstrate that converse and inverse may be false while contrapositive is equivalent to the original implication.

Examine the if and only if equivalence, formed by p implies q and q implies p, true only when p and q share the value, representing necessary and sufficient conditions.

Distinguish equivalence from equality by using the correct sign for statements versus numbers, and apply this to examples like the difference of squares and sign-based truth conditions.

Explore tautology as a logical formula, distinguish boolean expressions from algebraic formulas, and review basic laws with truth tables.

Learn to verify tautologies using truth tables and De Morgan's laws, with step-by-step proofs for p and q, and a method for building tables with n variables and k symbols.

Explore tautologies and implications by proving p implies q equals not p or q, then use Demorgan's law and double negation to obtain p and not q.

Demonstrate that the contrapositive of an implication is equivalent to the implication, while the converse and inverse are not, with tautologies and proofs using antecedent and consequent.

Explore a tautology proof of the distributivity of conjunction over disjunction using truth tables for p, q, and r, showing left and right sides are equivalent.

Construct a truth table for a logical expression with p, q, r, and negations to assess tautology status; conclude the expression is not always true, thus not a tautology.

Construct a truth table for p, q, r to test the given expression. The lecture demonstrates that p→q and q→r imply p→r, showing the expression is a tautology.

Apply tautologies to rewrite the set of points (x,y) with |x|+|y|<2 ⇒ x^2+y^2<1 as not P or Q. Sketch the unit disk and outside the square to show unbounded region.

Explore the existential quantifiers there exists, there exists exactly one, and for all, applying to open statements with domains like real numbers and natural numbers.

Explore universal quantifiers and the impact of order and precedence on statements, with real-number examples illustrating for all, exists, and the importance of quantifier placement.

Explore negations of quantifiers and De Morgan’s laws for one- and two-variable statements, and apply them to prove that f(x)=1/x is not decreasing on the real numbers excluding zero.

Practice solving logic problems by writing formal definitions for bounded functions, global and local extrema, and injective and surjective mappings, using quantifiers, domains, and contrapositive.

Explore translating plain English statements into mathematical symbols, including even numbers, sums of squares, primes, no largest natural number, and the non-negativity of real squares.

Present the statement that the sum of two positive integers is positive in logical symbols, using for all x, y in the positive integers and an implication.

Explain almost all natural numbers satisfy a property for the sequence a_n = 1/n and formalize that a_n tends to zero via epsilon neighborhoods, noting the finite exceptions.

Explore the epsilon-delta definition of a limit as x approaches a, using delta and epsilon to bound f(x) near L within a punctured neighborhood, with quantifiers and implications.

Learn to express that the limit is not L by negating the limit definition using epsilon-delta, quantifier changes, and the P and Q framework, with geometric intuition.

Explore primitive notions of sets—belonging to a set and the empty set—using roster and set-builder notation, intervals, and Venn diagrams.

Define a subset as all elements of C belonging to B, illustrate with a Venn diagram, and explain the universe as the maximal context for complements, varying by course.

Explore the correspondence between set theory and logic, using familiar symbols and statements, and see how complements, union, intersection, implication, and De Morgan's laws align across the two subjects.

Explore intersections of sets defined by conjunction, including domain determination for sums of functions, disjoint sets, empty set, and Venn diagrams, linking conjunction, intersection, and symbols.

Explore how the union of sets is defined by disjunction, including unions of infinitely many sets with existential quantifiers, and apply to domains and intervals in functions.

Show the axiom of extensionality: two sets are equal if and only if they share all elements, regardless of description, as with the empty set.

Define a subset via implication: for all x, if x is in A, then x is in B, and see how equality means subsets while negating shows not a subset.

Explore how the empty set yields a false statement, contrast with true statements, and use subset and implication logic to show the empty set is a subset of all sets.

Explore the concept of a true statement within a defined universe, showing that x belonging to the universe is always true in that context; understand context dependence and set notation.

Explore set difference, defined as elements in A not in B, illustrated with Venn diagrams and notation A minus B, and show that set difference is not commutative, with examples.

Introduce the symmetric difference of two sets as the union of their differences, with the form x in A and not B or x in B and not A.

Learn how to define the complement of a set via negation relative to a universe, with Venn-like illustrations and various examples.

Explore how the laws of logic and the laws of set theory align, with parallels between connectives and set operations, and preview proofs of De Morgan's laws.

Illustrates De Morgan's laws for sets with graphical diagrams and a formal proof, showing the equivalence of left and right sides through union, intersection, and complements.

Demonstrates the distributive law for sets, A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C), using union and intersection definitions and De Morgan's law.

Demonstrating how to prove theory laws with operations, this lecture derives a difference form and proves the symmetric difference formula A Δ B = (A ∪ B) \ (A ∩ B).

Define the Cartesian product of two sets as all ordered pairs with elements from each set, illustrated on the real plane R2, and extended to finite sets and intervals.

Explore the power set of a set, its subset definition, and basic boolean algebra operations like union, intersection, and complement, illustrated with two, three, and four element sets.

Explore the cardinality of sets by defining it as the number of elements in a set, and distinguish finite versus infinite cases with common notations such as card(A) and |A|.

Encode subset membership with zeros and ones across n elements. Each binary arrangement defines a unique subset, giving the power set's cardinality as 2^n.

Explore equinumerous sets and cardinal numbers through bijections, finite and infinite sets, including aleph zero and the continuum, with notes on their relevance to calculus.

Explain finite, countable, and uncountable sets through cardinalities, aleph-zero, and surjections from natural numbers; show how infinite countable sets form a sequence via injections and bijections.

Explore strange yet true facts about cardinalities of infinite sets by constructing surjections and bijections between natural numbers, even numbers, integers, and rational numbers via cartesian products to show countability.

Explore how any open interval, via a tangent-based bijection and a linear mapping, has the same cardinality as the real numbers, illustrating continuum-sized intervals.

Explore the inclusion-exclusion principle for finite sets, learn how to compute the cardinality of unions for two and three sets, and apply this method to problems like coffee and tea.

Apply the inclusion–exclusion principle to a three-language problem with English, Swedish, and Polish speakers, finding 12 speak all three languages and 11 speak only one.

Explore the transposition law for implications through subset-based, Venn diagram illustrations, showing that A ⊆ B iff B complement ⊆ A complement under a universal quantifier.

Explore the concept of relations, including equivalence and order relations, by linking everyday examples like siblings, parent–child, belonging, and genres, and connect them to mathematical relations.

Explore mathematical relations by mapping real-world ideas to parallelity and similarity, divisor relations, and inclusion and equality, and learn about transitivity and binary relations between two elements.

Explore binary relations as subsets of the Cartesian product X×X, learn notation and how to express x related to y, and define the domain and range of a relation.

Plot relations as subsets of the Cartesian product and visualize them on the plane, illustrating equality, less than, less than or equal, and greater than relations on real numbers.

Plot relations on a finite set as a subset of the cartesian product, using a table to mark 1 for related pairs and 0 otherwise; explore divisibility, squares, and parity.

Explore how to depict relations on finite sets with directed graphs, including nodes, edges, and loops, and visualize examples like less than, divisibility, and parity relations.

Explore equivalence relations, defined by reflexivity, symmetry, and transitivity, and how they create equivalence classes; see examples like parallelity and triangle similarity.

Determine if a relation on a finite set is an equivalence relation by examining plots and tables for reflexivity, symmetry, and transitivity. Use the diagonal to assess reflexivity and symmetry.

Learn to recognize equivalence relations on finite sets by analyzing graphs: verify reflexivity with loops at each vertex, symmetry with bidirectional edges, and transitivity via triangular closures.

Defines relation congruence modulo n as x and y sharing the same remainder when divided by n. Shows reflexivity, symmetry, and transitivity, with examples in modulo 2 and modulo 3.

Explore equivalence classes and partitions of integers under congruence modulo n, including quotient sets, representatives, and applications to vectors, parallel lines, and similar triangles.

Explore properties of binary relations, including irreflexive, antisymmetric, asymmetric, connected, and strongly connected relations, and apply them to order relations and divisibility on natural numbers.

Explore order relations by generalizing less than and related inequalities to non-strict and strict partial orders, noting reflexivity, antisymmetry, transitivity, and connectivity that yield total orders.

Explore partial orders and Hasse diagrams with two examples: proper subset relations on a 4-element set and proper divisors of 60, highlighting irreflexive, asymmetric, and transitive properties and diagram rules.

Explore relations on a finite set by testing reflexivity, symmetry, transitivity, antisymmetry, and connectivity. Solve problems on the set {A,B,C,D} with tables and graphs.

This lecture analyzes a two-part relation on natural numbers, showing reflexive, symmetric, and transitive properties, and concludes the relation is an equivalence relation on N.

Analyze a real-number relation defined by absolute value equations, visualized as two lines, and determine that it is not reflexive, not symmetric, not antisymmetric, not transitive, and not connected.

Relate X and Y if they contain the same even numbers. The relation is reflexive, symmetric, transitive, forming an equivalence relation, but not antisymmetric or connected.

Explore how functions are a special kind of relation, using graphs, domain and range, and the Cartesian product to compare graphs of both concepts.

Show that a function is a relation, but not every relation is a function. A function assigns a y to each x in its domain, and passes vertical line test.

Learn that every relation is invertible by swapping coordinates in each pair, yielding the inverse relation on the Cartesian product of y and x.

Analyze a relation from X to Y by identifying its departure set, target set, domain, range, and inverse, then determine if it is a function.

Analyze whether given relations are functions using the vertical line test, identify domain and range, and note non-injectivity or non-surjectivity while solving a problem set.

Explore relations and functions by constructing the Cartesian product of x and y, classify maps as surjective, injective, or bijective, determine inverses, and verify composition yields identity on x.

Verify equality of two functions by ensuring the same departure set and target set, identical domains, and equal values, using the quiz on f and g.

Explore surjections and injections through problem five, proving that h = g ∘ f is surjective if f and g are, and injective if both are, with counterexamples discussed.

Prove that an injective function is left cancelable: if f∘g1 = f∘g2, then g1 = g2 for any set X; illustrate with logarithm and exponential examples.

Prove a theorem about inverse functions by assuming g and h satisfy inverse relations with f, then show f is bijective and g and h are its inverses.

Explain the associativity of function composition and the invertibility of composites using inverses. Connect these ideas to linear algebra and calculus, with notes on relations generalizations.