3. Find a counter-example to these universally quanti?ed statements, where the domain for all variables consists of all integers.

1

a) ?x?y(y2 = x) b) ?x?y(xy = x)

c) ?x?y(x = 1/y) d) ?x?y(x3 6= y3)

Proofs

4. Prove that the additive inverse, or negative, of an even number is also an even number. (hint: you may use a direct proof)

5. Prove that min(a,min(b,c)) = min(min(a,b),c) whenever a,b, and c are real numbers. (hint: you may use a direct proof by cases)

6. Prove that if x and y are real numbers, then max(x,y) + min(x,y) = x + y. (hint: you may use a direct proof by cases)

7. Prove that for any real number r, if r2 is irrational, then r is irrational. (hint: you may use an indirect proof by contraposition)

8. Prove that the di?erence of any rational number and any irrational number is irrational. (hint: you may use an indirect proof by contradiction) 9. Bonus Question. Prove that v3 is irrational. (hint: one way to prove this statement is to use cases inside a proof by contradiction).