...9 Let A be a subset of {1, 2, 11} such that |A| = 7. Prove that there exist ï, y ¤ A such that ï ‡ y and x + y = 12.

Let S = {x | 6x – 1 E Q} and T be the - set of all odd integers. Prove that S and T have the same cardinality (no need to prove bijectivity of functions). Do not use the Schroder-Bernstein Theorem.

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?

Assume R1 and R2 are reflective relations on a set A. Prove or disprove each of the following statements. (a) R1 U R2 is reflexive. (b) Rị N R2 is reflexive.