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J 48 there is no way to express 3. Hence at least one of the statements p c ,q c has the truth value 1, so at least one of the statements 3x p x and 3x q x is true. Conversely, if 3x p x V 3x q x is true, then at least one of p a , q b has truth value 1, for some a, bin the prescribed universe.

Assume without loss of generality that it is p a. Then p a is true as is q a for all a in the universe, so the statements Vx p x , Vx q x are true.

Conversely, suppose that Vx p x A Vx q x is a true statement. Then Vx p x , Vx q x are both true. So now let c be any element in the prescribed universe. Suppose that the statement Vx p x V Vx q x is true, and suppose without loss of generality that Vx p x is true. This follows from Definition 2. Consequently, by Definition 2. Proof: Let us assume without loss of generality that k is even. Then kR. Consequently, kf is even - once again, by Definition 2. Proof: Let us assume without loss of generality that k is even and f is odd.

Then it follows from Definition 2. From Definition 2. So again by Definition 2. Proof: Assume that for some integer n, n2 is odd while n is not odd. Thus we have arrived at a contradiction since we now have n2 both odd at the start and even. This contradiction came about from the false assumption that n is not odd.

Here we must. Proof i : Using the method of contraposition, suppose that n is not even - that is, n is odd. Hence n2 is odd or, not even.

Since 2c2 is an integer, it follows that n2 is even. So by Definition 2. So we may writ. This then implies that n 2 b - 5 , for the integer b - 5. But with n both even as shown and odd as in the hypothesis we have arrived at a contradiction. Proof: [Here we provide a direct proof. Proof: We shall prove the given result by establishing the truth of its logically equivalent contrapositive.

The given result now follows by this indirect method of proof by the contrapositive. Proof: Since 4n is odd. It then follows from Definition 2. Consequently, the converse follows by contraposi tion.

Conversely, suppose that n is not even. Consequently, the converse follows by contraposition.

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## Discrete and Combinatorial Mathematics, 5th Edition

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