Suppose R and S are relations from A to B. Must the following statements be true? Justify your answers with proofs or counterexamples. a. R = Dom(R) × Ran(R) b. (R ∩ S)–1 = R–1 ∩ S–1

Prove: Let a,b,x and y be integers. If a|x and b|y then (ab)|(2xy).

If A is 3 x 3 and det (A) = 0, then select all of the following statements which are true:

When writing proofs by contradiction we begin by assuming the opposite of a statement, and then show that this leads to (i.e. entails) a contradiction. Which of the following statements constitutes a contradiction (that is, which of the following evaluates to false)? Suppose R(x, y) is a binary relation that relates two natural numbers. Select all that apply. A. O (3æ e N.R(x, æ) V R(x + 1, æ + 1)) ^ (Væ.¬R(x, æ)) B. O (Væ, y E N.R(x, y) → -R(y, x)) A R(3, 3) C. O (3r, y E N.R(x, y) → R(y, x)) ^ R(3, 4) A R(4, 3) D. O Va, y, z E N. (-(R(x,y) ^ R(y, z)) V R(x, z)) ) A R(2, 4) ^ R(4, 6) E. O None of the above