Mathematical proofs using, direct method, contarpositive methods, contradiction conditional and so o

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Mathematical proofs using, direct method, contarpositive methods, contradiction conditional and so on

 

Proofs to Complete 

Prove the following statements using an appropriate proof technique. 

I. Let a, b, c e Z. If alb and alc, thena2 1 bc 

2. Let n e Z. If 5n — 3 is even, then n is odd. 

3. 

Z. Then + n is even. 

4. Let n e Z. n is odd if and only if n2 is odd. (You may not use the fact that n is even 

if and only if n2 is even.) 

5. If a, b, c e Z and a2 + b2 c2 , then a is even or b is even. (Hint: you may use the 

fact that if n2 is even, then n is even.) 

1 

1 

2 

Prove or disprove the following statements: 

6. Let a, b, c Z. If a2 1 bc, then alb and alc. 

1 

7. For all e R such thatc O and 2 

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