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.

Determine whether the relation R on the set of all Web pages is reflexive, symmetric, antisymmetric, and/or transitive, where (a, b) ∈ R if and only if a) everyone who has visited Web page a has also visited Web page b. b) there are no common links found on both Web page a and Web page b. c) there is at least one common link on Web page a and Web page b. d) there is a Web page that includes links to both Web page a and Web page b.

Determine whether each relation defined on the set of positive integers is reflexive, symmetric, antisymmetric, transitive, and/or a partial order. $(x,y)\in R$ if $x=y$

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 given relation defined on the set of positive integers is reflexive, symmetric, anti symmetric or transitive. xRy if x-y =2

Suppose that the relation R is irreflexive. Is R² necessarily irreflexive? Give a reason for your answer.

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

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

Let R be a symmetric relation. Show that

$$R^n$$

is symmetric for all positive integers n.

Let R be a reflexive relation on a set A. Show that

$$R^n$$

is reflexive for all positive integers n.

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

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

$$a) R^{−1}.$$

b) R̅.

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

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.