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 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.

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

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

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 A be the set of students at your school and B the set of books in the school library. Let R₁ and R₂ be the relations consisting of all ordered pairs (a, b), where student a is required to read book b in a course, and where student a has read book b, respectively. Describe the ordered pairs in each of these relations. a) R₁ ∪ R₂ b) R₁ ∩ R₂ c) R₁ ⊕ R₂ d) R₁ − R₂ e) R₂ − R₁

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

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 .

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 R be the relation R = {(a, b) | a < b} on the set of integers. Find

$$a) R^{−1}.$$

b) R̅.

List the ordered pairs in the relation R from A = {0, 1, 2, 3, 4} to B = {0, 1, 2, 3), where (a, b) ∈ R if and only if a) a = b b) a + b = 4 c) a > b d) a | b e) gcd(a, b) = 1 f) lcm(a, b) = 2

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)?

a) List all the ordered pairs in the relation R = {(a, b) | a divides b} on the set {1, 2, 3, 4, 5, 6}. b) Display this relation graphically. c) Display this relation in tabular form.

Let R be the relation on the set of all states in the United States consisting of pairs (a, b) where state a borders state b. Find a) R⁻¹. b) R̅.