Let A be a nonempty set and R be a relation on A such that domain(R) = A. Prove that if R is symmetric and transitive then R is an equivalence.

(a) Define the following terms: (i) Cartesian product of two sets; (ii) relation on a set X. (b) Write down a relation on the set {1,2,3} which is reflexive and symmetric but not transitive. (c) Let S be the relation on the set R \ {0} defined by xSy if and only if y/x € Q. Prove that S is an equivalence relation.

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?

Prove or disprove the following statements

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 and S be reflexive relations over set A. Prove or disprove that RnS is reflexive

Let R be a relation and S its reverse. Show that R is injective if and only if S is well defined, and that R is surjective if and only if S is everywhere defined.

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.