Question

In: Computer Science

1. a) Prove that if n is an odd number then 3n + 1is an even...

1. a) Prove that if n is an odd number then 3n + 1is an even number. Use direct proof.

b) Prove that if n is an odd number then n^2+ 3 is divisible by 4. Use direct proof.

2. a) Prove that sum of an even number and an odd number is an odd number. Use direct proof.

b) Prove that product of two rational numbers is a rational number. Use direct proof.

3. a) Prove that if n2is an even number then n is an even number. Use indirect proof.

b) Prove that if n3is an odd number then n is an odd number. Use indirect proof

4. Prove that n2is an even number if and only if 3n + 1 is an odd number by proving

a) If n2is even then 3n + 1 is odd.

b) If 3n + 1 is odd then n^2 is even.

Expert Solution

ODD NUMBER : Any integer (not a fraction ) that can't be divided exactly by 2. E.g 1,3,5,7,9,11,...

EVEN NUMBER : Any integer that can be divided exactly by 2 is an even number . E.g : 2,4,6,8,10,12,...

1 (a) - Let odd number be n

n=1,3,5,7,...

3n+1=even (we've to proof)

putting value of n in above equation :-

3(1)+1=3+1=4 (4 is even)

3(3)+1=9+1=10 (10 is even)

3(5)+1=15+1=16 (16 is even)

1 (b) - Let odd number be n

n=1,3,5,7,9,...

n^2 +3= x (we've to proof that x is divisible by 4)

putting value of n in above equation:-

(1)^2+3=4 (4 is divisible by 4)

3^2 + 3 = 9 + 3 =12 (12 is divisible by 4)

5^2 + 3 = 25 + 3 =28 (28 is divisible by 4)

2 (a) : - Let even number be E = 2,4,6,8,10,12,...

Let odd number be O = 1,3,5,7,9,11,...

we've to proof Sum of even +odd =odd (E+O=O)

2+1=3, 2+3=5 , 2+ 5=7 ( 2 is even , 1,3,5, is odd and resultant 3,5,7 is odd)

4+1=5 , 4+3=7 , 4+5=9 ( 4 is even , 1,3,5 =odd , and resultant 1,7,9 is odd)

2 (b) : - Rational number : - A rational number can be defined as any number which can be represented in the form p/q where q is not equal to 0.The ratio p/q can be further simplified and represented in decimal form.

let two rational number be 3/4 and 3/5

now multiplying both 3/4 * 3/5 = 9/20 ( 9/20 is also a rational number)

venereology answered 5 months ago

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