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.
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