By Euler L.

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**Example text**

For each of the cases below, write a tautology using the given statement form. For example, if you are given P ∨ ¬Q, you might write (P ∨ ¬Q) ↔ (Q → P). (a) ¬(¬P); (b) ¬(P ∨ Q); (c) ¬(P ∧ Q); 2 Logically Speaking 23 (d) P → Q. 12. When we write, we should make certain that we say what we mean. If we write P ∧ Q ∨ R, you may be confused, since we haven’t said what to do when you are given a conjunction followed by a disjunction. Put parentheses in to create a statement form with the given truth table.

Describe your method of proof. 19. ” (a) State the contrapositive of this implication. (b) State the converse of this implication. For parts (c) and (d), assume the original statement is true. (c) Suppose someone tells you that Simon did not take German. What, if anything, can you conclude about Simon? Why? (d) Suppose someone tells you that Simon took French. What, if anything, can you conclude about Simon? Why? 20. Consider the statement form (P ∨ ¬Q) → (R ∧ Q). (a) Write out the truth table for this form.

16. Let x and y be real numbers. Show that if x = y and x, y ≥ 0, then x2 = y2 . 17. In the statement below, G is a group and H is a normal subgroup of G. ” (a) State the converse of this statement. (b) State the contrapositive of this statement. ” Write this in terms of your answers to the first two parts of this problem. 18. Prove that if the product of two integers x and y is odd, then both integers are odd. Describe your method of proof. 19. ” (a) State the contrapositive of this implication.