Let R be the relation on the set {0, 1, 2, 3} containing the ordered pairs (0, 1), (1, 1), (1, 2), (2, 0), (2, 2), and (3, 0). Find the a) reflexive closure of R. b) symmetric closure of R.

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

$$a) R^2. b) R^3. c) R^4. d) R^5.$$

R. S. and T are relations defined on A={0, 1, 2, 3}. Let R={(0, 1), (0, 2), (1, 1), (1, 3), (2, 2), (3, 0)}. Find R′, the transitive closure of R.

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.

How can the directed graph representing the symmetric closure of a relation on a finite set be constructed from the directed graph for this relation?

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.

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

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

In number of relations are defined on the set A={0, 1, 2, 3}. For each relation: a. Draw the directed graph. b. Determine whether the relation is reflexive. c. Determine whether the relation is symmetric. d. Determine whether the relation is transitive. Give a counterexample in each case in which the relation does not satisfy one of the properties.

$$R _ { 8 } = \{ ( 0,0 ) , ( 1,1 ) \}$$

(a) Let $C \subseteq [ 0,1 ]$ be uncountable. Show that there exists $a \in ( 0,1 )$ such that $C \cap [ a , 1 ]$ is uncountable. (b) Now let $A$ be the set of all $a \in ( 0,1 )$ such that $C \cap [ a , 1 ]$ is uncountable, and set $\alpha = \sup A .$ Is $C \cap [ \alpha , 1 ]$ an uncountable set? (c) Does the statement in (a) remain true if "uncountable" is replaced by "infinite"?

Let X = {0, 1}. (a) List (as subsets of $X \times X)$ all possible relations on X. (b) Which of the relations in part (a) are equivalence relations?

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

$$R^n$$

is reflexive for all positive integers n.