. Prove the following statements using direct proof.[10 points each (Remember th

Question

. Prove the following statements using direct proof.[10 points each (Remember this course is about learning how to read/write formal proofs! Use formal definitions in your proofs all the time including the formal definitions of odd and even numbers! "Odd times odd is odd", "odd+odd-even" type of arguments are not part of our formal proofs! (a) Let a be an odd integer. Use formal definition of odd integers to prove that a2 +2a1 is even Note that following is NOT a proof of this statement "Let a-3, then 32 +2(3) +1-9+6+1 16 is even." (b) Suppose a, b, ce Z. Prove that if a2|& and b3k, then an (c) Prove by using the cases that if e R, then r +3-x> 2

Explanation / Answer

a)

Given a is odd integer let a=2k+1

a2+2a+1 = (2k+1)2+2(k+1)+1

=2k2+1+4k+2k+2+1

=2k2 + 6k+2

=2(k2+3k+1) =even

because when 2 is multiplied with any(odd or even) number we always get even number

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b)

a2 | b means b= a2*k where k is quoient

b3 | c means c= b3 *p where p is quoient

c= b3 *p

c= (a2*k)3 *p

c= a6*k3*p

dividend= =divisor * quoient

c6 = k3*q

Hence a6 | c

Hence proved

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c) |x+3|-x>2

case 1: if x<3

-(x+3)-x>2

-x-3-x>2

-2x-3>2

-2x>5

-x>5/2

x<-2.5

not possible because x<3

second case

x>=3

x+3-x>2

3>2

it is true

so x belongs from 3 to infinity.

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