Let R be a relation on a nonempty set A such that IA CR, R = R¹ and RoRCR. Prove explicitly that R is an equivalence relation on A.

Let R be the relation defined on P({1,., 100}) by ARB if and only if |A n B| is even. Is R reflexive? Is R symmetric? Is R anti-symmetric? Is R transitive?

Let T be the set {w = {0, 1}* ||w| ≤ 4}. Let R be the equivalence relation defined on T as follows: R = {(x, y) | x ≤T, yɛT, no(x) = = no(y)}, where no(r) represents the number of zeroes in the string x, and no(y) represents the number of zeroes in the string y. For example, (1011, 01) is a pair in R because the two strings 1011 and 01 have the same number of zeroes as each other. Every element in the set will appear in exactly one equivalence class and will be related to all elements in its class and not related to any elements outside of its class. What are the equivalence classes of T created by the relation R?

4. Let R be a relation defined on Z as follows: For all m, n € Z, m R n iff 4 | (m² — n²). a) Prove that R is an equivalence relation. b) Describe the distinct equivalence classes of the relation R. c) Do the distinct equivalence classes form a partition of Z? Explain.

Let X be the set of nonempty sets of {-1, 0, 1} and define a relation R on X as follows: For all sets s and tin X, s R t⇒ the sum of the elements in s equals the sum of the elements in t. It is a fact that R is an equivalence relation on X. Use set-roster notation to list the distinct equivalence classes of R. (Enter your answer as a comma-separated list of sets. Enter EMPTY or Ø for the empty set.)

Let R be the relation on Z defined by x Ry if and only if x+ 3y is even. Prove that R is an equivalence relation.

Let R and S be equivalence relations on a set A; recall that by definitionR, S ⊆ A × A. Prove that R ∩ S is an equivalence relation.

8. Let A be a set of nonzero integers and let R be a relation on A × A defined by (a, b)R(c, d) whenever ad = bc. Show that R is an equivalence relation. That is, R is reflexive, symmetric, and transitive.