List the 16 different relations on the set {0, 1}.

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.

For each of these relations on the set {1, 2, 3, 4), decide whether it is reflexive, whether it is symmetric, whether it is antisymmetric, and whether it is transitive. a) {(2, 2), (2, 3), (2, 4), (3, 2), (3, 3), (3, 4)} b) {(1, 1), (1, 2), (2, 1), (2, 2), (3, 3), (4, 4)) c) {(2, 4), (4, 2) d) {(1, 2), (2, 3), (3, 4)} e) {(1, 1), (2, 2), (3, 3), (4, 4)} f) {(1, 3), (1, 4), (2, 3), (2, 4), (3, 1), (3, 4)}

Let R be the relation {(1, 2), (1, 3), (2, 3), (2, 4), (3, 1)}, and let S be the relation {(2, 1), (3, 1), (3, 2), (4, 2)}.

Find S ◦ R.

Let S be a set with n elements and let a and b be distinct elements of S. How many relations R are there on S such that a) (a, b) ∈ R? b) (a, b) ∉ R? c) no ordered pair in R has a as its first element? d) at least one ordered pair in R has a as its first element? e) no ordered pair in R has a as its first element or b as its second element? f) at least one ordered pair in R either has a as its first element or has b as its second element?

Give an example of a relation on a set that is a) both symmetric and antisymmetric. b) neither symmetric nor antisymmetric.

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

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

Let R₁ and R₂ be the “divides” and “is a multiple of” relations on the set of all positive integers, respectively. That is, R₁ = {(a, b) | a divides b} and R₂ = {(a, b) | a is a multiple of b}. Find a) R₁ ∪ R₂. b) R₁ ∩ R₂. c) R₁ − R₂. d) R₂ − R₁. e) R₁ ⊕ R₂.

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.

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.

Let R₁ and R₂ be the “congruent modulo 3” and the “congruent modulo 4” relations, respectively, on the set of integers. That is, R₁ = {(a, b) | a ≡ b (mod 3)} and R₂ = {(a, b) | a ≡ b (mod 4)}. Find a) R₁ ∪ R₂. b) R₁ ∩ R₂. c) R₁ − R₂. d) R₂ − R₁. e) R₁ ⊕ R₂.

Let R be a symmetric relation. Show that

$$R^n$$

is symmetric for all positive integers n.

What is the probability that when a coin is flipped six times in a row, it lands heads up every time?

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)?