(2) Show that for any integers a,b we always have a? – 4b # 2. -

Consider the proposition ((x → (y v z)) → (x → y) v (x → z) a. Is it a tautology or a contradiction? b. Prove your answer for item a. without using a truth table.

Prove the following statements;

5. Write the following statements symbolically and then demonstrate whether they are true or false. If true, then prove it, If false, then provide a counterexample or prove that the negation of the statement is true. (a) For all integers, if 9 divides n2, then 9 divides n. (b) There exist integers m and n such that 15m + 12n = -6.

The following arguments have false conclusions. What is the problem with each argument? A. If a person owns a cat, then they don't own a dog. Dr. Morales owns a cat. :. Dr. Morales doesn't own a dog B. If a number n is even, then 2n is even. The number 2n is even, for n=3 (note: 2n = 6). : The number n is even, for n=3.

Construct direct proofs to show that the following symbolic arguments are valid. Commas mark the breaks between premises. ~C→(F→C), ~C ∴ ~F

Write the negation of of the following statement: “For every pair of real numbers x and y if x < y, then there exists a rational number q such that x < q < y.”