Discrete Math | Learn Discrete Mathematics by easy methods

Explore set operations in discrete math, including unions and intersections of B and C and the cross product A × (B ∩ C), with step-by-step examples.

Compute the cross products A × B and B × A for A = {1,2,3} and B = {2,3}, listing all ordered pairs like (1,2), (1,3), (2,2), (2,3), (3,2), (3,3).

We define a relation on the natural numbers via a+3b=12, derive the ordered pairs (9,1), (6,2), (3,3), and identify the domain {9,6,3} and range {1,2,3}.

Illustration2 shows how R1, R2, and R3 are relations from A to B, each a subset of A × B, while R4 fails because 4 is not in A.

Explore the less-than relation on sets A and B by listing all ordered pairs, forming R, and computing its inverse R inverse, with concrete pairings such as (1,4), (1,5), (4,5).

Learn how even functions satisfy f(-x) = f(x) and odd functions satisfy f(-x) = -f(x), and identify functions that are neither even nor odd.

Explore domain determination and function evaluation, including explicit and implicit expressions, plus tests for even and odd behavior, using logarithm and rational function properties.

Understand composite functions: g∘f maps A to C as g(f(x)) with f:A→B and g:B→C, and may differ from f∘g, yet is associative and bijective when f and g are bijective.

Analyze function composition with f∘f(x) = x^3 and g∘g(x) = 3x-1, then compute f∘g(x) = (3x-1)^3 and g∘f(x) = 3x^3-1, showing they differ.

Explore the composition of functions with g(f(x)) and f(g(x)) using f(x)=x^2+2 and g(x)=x/(x-1), showing that the two compositions need not be equal.

Compute f(f(x)) where f(x) = (x-1)/(x+1) under the domain x != -1 and x != 0. Show that f(f(x)) = -1/x by simplifying the nested expression and canceling terms.

Show that the function f(x) = (3x-2)/(2x-3) is an involution, since f(f(x)) = x for all real x not equal to 3/2.

Illustrates composing the function f(x)=x/√(1+x^2) through repeated applications and proves that f(f(f(x))) = x/√(1+3x^2) via step-by-step algebraic simplification.

Derive the inverse of f x equals 2x minus 3 by solving for x in terms of y, then replace y by x to obtain f inverse x equals (x+3)/2.

Find the value of f inverse nine by setting f inverse nine equal to x, which yields f x equals nine, and solving x^2+5x=0 to obtain x=0 or x=-5.

Find the inverse of f(x) = 1/(1 − x) by solving for x in terms of y, yielding f^{-1}(x) = (x − 1)/x.

Illustration presents basic counting and sequence ideas from one to nine, forming chains and groups of even terms while introducing simple numerical reasoning in discrete math.

Illustration 2 presents ways to rewrite and simplify numeric expressions involving 11, including x over 11, 11 x 11, and 121, with examples.

Shane Watson explains factorials, highlighting 4!, 5!, and 6!, and shows that 6! equals 720.

Explore discrete math concepts using factorials and simple algebra with numbers, illustrated by a pictorial approach to a plus two and a plus one.

Explore the fundamental principle of counting and the multiplication rule, showing how the number of outcomes equals the product of choices in sequence, with examples of selecting students and options.

Illustration 1 demonstrates the multiplication principle in discrete math by counting sequential choices: eight options for the first activity and seven for the second, yielding 56 possible complete sequences.

Learn to count four-letter words under word rules, comparing repetition allowed versus not allowed. Apply the multiplication principle and permutation concepts to compute the counts for each case.

Understand permutations, where order matters, and apply the factorial formula to count arrangements of objects.

Explore permutations using the nPr formula and factorials, illustrating counting arrangements of letters and objects, including word formation and comparison of different arrangement cases.

Learn how combinations capture the number of ways to choose objects when order does not matter, using the n choose r formula and related identities.

Explore combinations through the ncr formula, learn to compute binomial coefficients, and apply methods with practical examples to solve counting problems in discrete math.

Study basic logic: define statements, distinguish simple from compound statements, and apply negation, conjunction, disjunction, and exclusive or with truth tables.

Construct a truth table for the negation of B, analyze cases where B is true or false, and determine how compound sentences evaluate to true or false.

Illustration 2 constructs a truth table for B or negation of A, examining true and false values and how negation affects the outcomes.

Explore the basic concepts of mathematical induction, including statements versus sentences, and demonstrate a first-principles induction proof using base and step cases on natural numbers.

Prove by the principle of mathematical induction that the sum of squares from 1 to n equals n(n+1)(2n+1)/6, with base case n = 1 and P(k) implies P(k+1).

Apply the principle of mathematical induction to prove the statement that the sum 1+2+...+n equals n(n+1)/2, establishing the base case and the inductive step from k to k+1.

Use the principle of mathematical induction to prove that 1 + x + x^2 + ... + x^n = (x^{n+1}-1)/(x-1), validating base case and inductive step.