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## 12. Let A = {A,, : ne Z, where A, = (n- 1,n+ 1).
(a) Find A and prove your conjecture.
(b) Find R-UA and prove your conjecture.
13. Prove the generalization, to an arbitrary collection of sets, of the DeMorgan's law
14. Prove : if A, B are two families of sets such that AC B, then JAC JB.
15. Prove : if DCA for all A E A, then DC A.
16. Prove : if A CD for all A E A, then JAC D.
17. Prove or find a counterexample for the conjecture : if An B = AnC, then B = C.
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18. Prove or find a counterexample for the conjecture : if AU B = An B, the A = B.
19. Prove or find a counterexample for the conjecture : if A, B are two families of sets such that
UA =UB, then A = B.
20. Prove that (A x C) U (B x D) C (AU B) x (CU D). Give a counterexample to show that the
reverse inclusion is not true.
21. Prove that A x (BnC) = (A × B) n (A × C).
22. Prove that (AUB) x C = (A x C)U(B × C).
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23. Supply a proof or a counterexample for each conjecture:
(a) Ax C = Bx C implies A = B.
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(b) Ax C = Bx C and C #0 implies A = B.
24. Prove or disprove (A x C) n (B × D) = (ANB) × (Cn D).