a) How many relations are there on the set {a, b, c, d}? b) How many relations are there on the set {a, b, c, d} that contain the pair (a, a)?

How many of the 16 different relations on {0, 1} contain the pair (0, 1)?

Let R be the relation on the set of people consisting of pairs (a, b), where a is a parent of b. Let S be the relation on the set of people consisting of pairs (a, b), where a and b are siblings (brothers or sisters). What are S ◦ R and R ◦ S?

Determine whether the relation R on the set of all people is reflexive, symmetric, antisymmetric, and/or transitive, where (a, b) ∈ R if and only if

a) a is taller than b.

b) a and b were born on the same day.

c) a has the same first name as b.

d) a and b have a common grandparent.

Represent each of these relations on {1, 2, 3, 4} with a matrix (with the elements of this set listed in increasing order). a) {(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)} b) {(1, 1), (1, 4), (2, 2), (3, 3), (4, 1)} c) {(1, 2), (1, 3), (1, 4), (2, 1), (2, 3), (2, 4), (3, 1), (3, 2), (3, 4), (4, 1), (4, 2), (4, 3)} d) {(2, 4), (3, 1), (3, 2), (3, 4)}

Which of the 16 relations on {0, 1} are a) reflexive? b) irreflexive?
c) symmetric? d) antisymmetric? e) asymmetric? f) transitive?

Deal with these relations on the set of real numbers: R₁ = {(a, b) ∈ R² | a > b}, the “greater than” relation, R₂ = {(a, b) ∈ R² | a ≥ b}, the “greater than or equal to” relation, R₃ = {(a, b) ∈ R² | a < b}, the “less than” relation, R₄ = {(a, b) ∈ R² | a ≤ b}, the “less than or equal to” relation, R₅ = {(a, b) ∈ R² | a = b}, the “equal to” relation, R₆ = {(a, b) ∈ R² | a ≠ b}, the “unequal to” relation. Find

$$a) R_1 ◦ R_1. b) R_1 ◦ R_2. c) R_1 ◦ R_3. d) R_1 ◦ R_4. e) R_1 ◦ R_5. f) R_1 ◦ R_6. g) R_2 ◦ R_3. h) R_3 ◦ R_3.$$

Determine whether the relation R on the set of all real numbers is reflexive, symmetric, antisymmetric, and/or transitive, where (x, y) ∈ R if and only if a) x + y = 0. b) x = ± y. c) x - y is a rational number. d) x = 2y. e) xy ≥ 0. f) xy = 0. g) x = 1. h) x = 1 or y = 1.

What is the probability that a five-card poker hand does not contain the queen of hearts?

Let R₁ = {(1, 2), (2, 3), (3, 4)} and R₂ = {(1, 1), (1, 2), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3), (3, 4)} be relations from {1, 2, 3} to {1, 2, 3, 4}. Find a) R₁ ∪ R₂. b) R₁ ∩ R₂. c) R₁ − R₂. d) R₂ − R₁.

Let R be the relation on the set {1, 2, 3, 4, 5} containing the ordered pairs (1, 3), (2, 4), (3, 1), (3, 5), (4, 3), (5, 1), (5, 2), and (5, 4). Find R^2 , R^3 , R^4 , R^5 , and R^6 .

How many relations are there on a set with n elements that are a) symmetric? b) antisymmetric? c) asymmetric? d) irreflexive? e) reflexive and symmetric? f) neither reflexive nor irreflexive?

a) Define the power set of a set S.

b) When is the empty set in the power set of a set S?

c) How many elements does the power set of a set $S$ with $n$ elements have?

Determine whether the relation R on the set of all integers is reflexive, symmetric, antisymmetric, and/or transitive, where (x, y) ∈ R if and only if a) x ≠ y. b) xy ≥ 1. c) x = y + 1 or x = y − 1. d) x ≡ y (mod 7). e) x is a multiple of y. f ) x and y are both negative or both nonnegative. g) x = y². h) x ≥ y².