Are the following implications correct? If yes, prove it. Otherwise, find counterexample. (a) (A⇒C) ⇒ (AVC ⇒Bv C) (b) (A⇒ BVC) ⇒ (A⇒ C) (c)¬AAA⇒ B

2. Use the Equational-style to prove the following: a. b. HAVAAB = A c. d. HAVB=AV¬B=A A V (B → A) = B → A A → B = ¬B¬A

which part is A(1) and which is A(2)?

(b) If r23= 1, prove that 2 R₁(23) = 12 = 13 and 123= 01 (1-12).

If A is nonsingular, 1 (a) A det A which of the following equals adj(A)? 1 (b) (det A) A-¹ det A (c) A-1 (d) (det A) A

13

(p v q) → r= O (p v r)A(-q v r) O (-p v r)v(q v r) O (-p v r)v(-q v r) O (-p v r)A(¬q v r)

The following are equivalent statements: (i) a∈R is constructible from Q (ii) (a,0) is constructible from Q×Q (iii) (a,a) is constructible from Q×Q (iv) (0,a) is constructible from Q×Q.

Is it true that det(A + B) = det(A) + det(B)? If this is so, explain why. If it is not so, give a counter example.

Let det(A) = 5 and det(A + B) = 20. We conclude that det(/+A-B) = 15 O True O False