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Prove that the relation a ≡ b (mod m) is an equivalence relation on the set of integers.
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- Let be a relation defined on the set of all integers by if and only if sum of and is odd. Decide whether or not is an equivalence relation. Justify your decision.True or False Label each of the following statements as either true or false. Let be an equivalence relation on a nonempty setand let and be in. If, then.Label each of the following statements as either true or false. If R is an equivalence relation on a nonempty set A, then any two equivalence classes of R contain the same number of element.
- 21. A relation on a nonempty set is called irreflexive if for all. Which of the relations in Exercise 2 are irreflexive? 2. In each of the following parts, a relation is defined on the set of all integers. Determine in each case whether or not is reflexive, symmetric, or transitive. Justify your answers. a. if and only if b. if and only if c. if and only if for some in . d. if and only if e. if and only if f. if and only if g. if and only if h. if and only if i. if and only if j. if and only if. k. if and only if.In Exercises 610, a relation R is defined on the set Z of all integers. In each case, prove that R is an equivalence relation. Find the distinct equivalence classes of R and list at least four members of each. xRy if and only if x+3y is a multiple of 4.Give an example of a relation R on a nonempty set A that is symmetric and transitive, but not reflexive.
- 4. Let be the relation “congruence modulo 5” defined on as follows: is congruent to modulo if and only if is a multiple of , and we write . a. Prove that “congruence modulo ” is an equivalence relation. b. List five members of each of the equivalence classes and .True or False Label each of the following statements as either true or false. If is an equivalence relation on a nonempty set, then the distinct equivalence classes of form a partition of.