Discrete Mathematics - Sets, Relations & Functions

Define a relation from A to B as a subset of A×B, the product set, and use roster form and arrow diagrams to illustrate domain and range.

Explore the inverse of a relation by swapping the domain and range, illustrated with an example and its arrow diagram to show how the inverse connects elements differently.

Explore the types of relations, including void, universal, and identity relations, and learn to express them in set-builder form, roster form, and arrow diagrams.

Examine reflexive, symmetric, and transitive relations on sets, using subsets of E×E. See how reflexive relations require each element to relate to itself, illustrated with R1 and R2 on {1,2,3}.

Explore symmetric relations, where a related to b implies b related to a, with reflexive and transitive properties illustrated. See how inverse equality signals symmetry in relations on a set.

Explore relations in discrete mathematics, including reflexive, symmetric, and transitive types, with examples like the real numbers order ≤, divisibility, and x is a brother of y.

Explore the concept of functions as a rule linking an independent variable to a dependent variable, with real-world applications including circumference, tax, debt, painting, tiling, and compound interest.

Explore the four interval types in mathematics: open (a,b), closed [a,b], open-closed (a,b], and closed-open [a,b), illustrating how endpoints are included or excluded.

Explore constant, identity, and exponential functions and their graphs. The video covers constant functions y=c, identity y=x, and exponential behavior with domain all real numbers and range (0, ∞).

Explains polynomial, rational, and irrational functions, their forms, and domains, noting polynomials are continuous, rational functions exclude denominator zeros, and irrational functions involve cube roots and similar non-integer values.

Explore the absolute value (modulus) function, its domain, range, and graph, plus key properties and the sign function, with examples showing x, ±x, and basic inequalities.

Learn the greatest integer (floor) function, which maps x to the largest integer less than or equal to x, yielding a step function with integer outputs on the real line.

Examine even and odd functions via f(-x) behavior, with examples like x, which is odd, and x^2 or modulus, which are even, and discuss product and derivative properties.

Understand identical functions: require equal domains and equal ranges, and that F(x) = G(x) for every x in the domain, for real-valued functions from A to B.

Explore one-to-one functions: injectivity tests, linear functions as always injective, and counting injective mappings from an n-element set to an m-element set.

Explore many-to-one functions, where multiple elements of A map to the same B, contrasted with one-to-one functions via the horizontal line test and even-degree polynomial examples.

Explain onto and into functions by showing when every codomain element has a preimage, and when some codomain elements lack preimages, using domain, range, and codomain comparisons.

Explains that even degree functions from real numbers to real numbers are not onto, while odd degree functions are onto, and distinguishes onto, into, and bijective functions with inverses.

Determine the domain of a function by identifying the x-values where the expression is defined, a subset of real numbers, with even roots requiring nonnegative radicands and denominators nonzero.

Explore determining the domain when combining two functions, including addition, subtraction, multiplication, and division, by taking the intersection of their domains and excluding division by zero.

Learn to determine the range of a function from its domain using y = f(x). Explore finite and infinite domains and when to express x as a function of y.

Explore the properties of logarithmic functions, including log_b(xy) = log_b x + log_b y, log_b(x/y) = log_b x - log_b y, log_b(a^m) = m log_b a, and change of base.

Explore the properties of the greatest integer function, including floor(-n) = -floor(n), floor(x+k) = floor(x) + k for integers k, and the behavior of floor(x) + floor(-x) depending on whether x is an integer.

Explore composite functions by composing f and g to form h, examine domains and codomains, and determine when the composition is well defined.

Explore the existence conditions for f∘g and g∘f, and how even/even, odd/odd, and mixed parity of f and g affect the composition's parity.

Explore the inverse function, defined when a function is one-to-one and onto, enabling a unique mapping from B back to A via its inverse.

Inverse functions are unique; the inverse of a composition F∘G is G^{-1}∘F^{-1}, and F∘G's inverse yields identity under proper domain and codomain.

Determine the number of students taking exactly one subject in a class of 55 using the three-set formula for math, physics, chemistry; intersections and triple 4 lead to 22.

Apply set operations to a class of 100: total math 55 and physics 67, find the intersection, then subtract from physics to obtain 45 who passed only physics.

Use Venn diagrams to evaluate E minus B and B minus E, then unite them to obtain the symmetric difference, i.e., (E ∪ B) minus (E ∩ B).

Explore numericals on sets by counting non-empty subsets of a four-element set, using the 2^n minus 1 formula and verifying with subset enumeration.

Explore numericals on sets using Venn diagrams, focusing on set difference and intersection with examples like E minus B and E intersection B to reinforce basic set identities.

Apply the union formula to sets of transport choices, compute the union of car or bus travelers as 20% + 50% - 10% = 60%, clarifying set relations.

Calculate |E ∪ B| using |E| + |B| − |E ∩ B|, then get the complement from the universal set: 20 − |E ∪ B|; the values give 17 and 3.

Demonstrates applying the distributive law to set operations—complements, unions, and intersections—in a level-one set theory numericals problem, yielding six.

Compute A ∩ B^c within the universal set of 1 to 10. With A = {1,2,5} and B = {6,7}, B^c = {1,2,3,4,5,8,9,10}, so A ∩ B^c = {1,2,5}.

Explore level-1 function numericals, solving logarithmic equations, change-of-base, quadratics, inverse and composite functions, and domain and range and greatest-integer and fractional-part problems, with step-by-step explanations.

Explore solved numericals on functions, including domain and range analyses, inverses and composition, and transformations across logs, polynomials, and trigonometric scenarios.

Explore solved numerical problems on functions, covering domain and range, one-to-one and inverse properties, monotonicity, and basic graphing techniques for level-3 questions.