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.

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 A is reflexive, irreflexive, symmetric, asymmetric, antisymmetric, or transitive.

$A$ is the set of all people in the world. $a R b$ if and only if $a$ is the sister of $b$.

For each of the following relations on the set of human beings, please determine whether the relation is reflexive, irreflexive, symmetric, antisymmetric, and/or transitive. a. has-the-same-last-name-as. b. is-the-child-of. c. has-the-same-parents-as (i.e., same mother and father). d. d. has-a-common-parent-as (i.e., same mother or father). e. is-married-to. f. is-an-ancestor-of.

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

$$R^n$$

is reflexive for all positive integers n.

Which of these relations on the set of all people are equivalence relations? Determine the properties of an equivalence relation that the others lack.

a) {(a, b) | a and b are the same age}

b) {(a, b) | a and b have the same parents}

c) {(a, b) | a and b share a common parent}

d) {(a, b) | a and b have met} e) {(a, b) | a and b speak a common language}

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

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.

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.

Determine whether the relation R on the set A is reflexive, irreflexive, symmetric, asymmetric, antisymmetric, or transitive.

$A$ is the set of all ordered pairs of real numbers; $(a, b) R(c, d)$ if and only if $a=c$.

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)?

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 given relation is an equivalence relation on the set of all people. {(x,y)|x and y are the same height}

Let R be a symmetric relation. Show that

$$R^n$$

is symmetric for all positive integers n.