We can easily prove this by counter example. is not an element of Q,but its square, 2 is an element of Q, contradicting the given statement. Hence disproved.
z 또 Q then re Q Exercise 10. Prove or disprove: If z 또 Q then re Q Exercise 10. Prove or disprove: If
Either prove or disprove (by giving a counterexample) this claim re- garding powersets and set containment: For any sets A and B, if A ⊆ B, then P(A) ⊆ P(B)
Prove or disprove each statement. 6. Vp € Q (3q Q (p+9 = 73)).
Prove or disprove the following equivalence claim. (r ∧ s ∨ ¬t) ⇒ q ≡ ( ¬r ∧ t) ∨ (¬s ∧ t) ∨ q
Prove or disprove (without using a truth table): (p^q) rightarrow (q rightarrow p) is a tautology. Prove that the contrapositive holds (without using a truth table), that is that the followi holds: p rightarrow q identicalto q rightarrow p
10. Let A, B, and C be sets. (a) Prove or disprove: if A - C CB-C, then ACB. (b) State the converse of part (a) and prove or disprove.
Exercise 2. Prove or disprove the following: a) C € P(A) + CCA b) ACB + P(A) CP(B) c) A=0 + P(A) = 0
1. Let a, b,cE Z be positive integers. Prove or disprove each of the following (a) If b | c, then gcd(a, b) gcd(a, c). (b) If b c, then ged(a., b) < gcd(a, c)
3. Prove the statements that are true and give counterexamples to disprove those that are false. (a). Va,b,n E Z* , if a’ =b}(modn) then a =b(modn). (8 points) (b). If p> 2 and q> 2 are prime, then p? +q must be composite. (12 points)
discrete math question using proofs to determine to prove the following equation or disprove it 4. Prove or disprove. Let A, B, C, and D be sets. Then (Ax B)n (CxD) (Ancx (B nD) 5. Prove or disprove: {2k 1 k E Q} {4" | k E Q) F6 7 Prove or disprove. Let A be a set and let I be an arbitrary index set for a collection of sets {Be l α E 1). Then, 6. An(UP)-a αΕΙ
Prove or Disprove #3 (d) For each of the following, prove or disprove: iii) There is an element of X × Y with the form (a, 3a) (d) For each of the following, prove or disprove: iii) There is an element of X × Y with the form (a, 3a)