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, SR 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 for the empty set.)

Prove or disprove the following statements

Let R be the following relation on the set of all people. R= {(a, b) : a and b share a common biological parent} Show whether or not R is an equivalence relation.

Let X = {−1, 0, 1} and A = ?(x) and define a relation R on A as follows: For all sets s and t in ?(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 A. 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 ∅ for the empty set.)

Let S be a nonempty set. Consider the relation ~ on P(S) defined by A ~ B if AUBC = S. Is the relation an equivalence relation? Explain. If yes, identify (no need to justify) the equivalence class of

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.