Solve the following situations as they pertain to relations: a.  Congruence modulo 5 is a relation on the set A = the set of integers (Z). In this relation, xRy means x is congruent to y(mod5). Write out the set R in set-builder notation. b.  Define a relation on the set of integers (Z) as xRy if |x - y| < 1. Is R reflexive? Symmetric? Transitive? If a property does not hold, say why. What familiar relation is this? c. Define a relation R on the set of integers (Z) by declaring that xRy if and only if x^2 is congruent to y^2 (mod 4). Prove that R is reflexive, symmetric, and transitive. d. Define a relation R on the set of integers (Z) as xRy if and only if x^2 + y^2 is even. Prove R is an equivalence relation. Describe its equivalence classes. e. Prove or disprove: If R and S are two equivalence relations on a set A, then R U S is also an equivalence relation on A.

Solve the following situations as they pertain to relations:

a.  Congruence modulo 5 is a relation on the set A = the set of integers (Z). In this relation, xRy means x is congruent to y(mod5). Write out the set R in set-builder notation.

b.  Define a relation on the set of integers (Z) as xRy if |x - y| < 1. Is R reflexive? Symmetric? Transitive?
 If a property does not hold, say why. What familiar relation is this?

c. Define a relation R on the set of integers (Z) by declaring that xRy if and only if x^2 is congruent to y^2 (mod 4). Prove that R is reflexive, symmetric, and transitive.

d. Define a relation R on the set of integers (Z) as xRy if and only if x^2 + y^2 is even. Prove R is an equivalence relation. Describe its equivalence classes.

e. Prove or disprove: If R and S are two equivalence relations on a set A, then R U S is also an equivalence relation on A.