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 4 months ago

Prove that every natural number is odd or even.

Prove the following theorem: Theorem ∀n ∈ Z, n is either even or odd (but not...

Prove the following theorem: Theorem ∀n ∈ Z, n is either even or odd (but not both). Your proof must address the following points: 1. n is even or odd (and nothing else). 2. n is odd =⇒ n is not even (hint: contradiction). 3. n is even=⇒ n is not odd (hint: contrapositive). The first point is a bit more difficult. Start by making a statement about 0. Then assuming that n is even, what can you say about...

prove that if an even integer n is subtracted from an odd integer m. then m...

prove that if an even integer n is subtracted from an odd integer m. then m - n is odd.

3. To begin a proof by contradiction for “If n is even then n+1 is odd,”...

3. To begin a proof by contradiction for “If n is even then n+1 is odd,” what would you “assume true? 4. Prove that the following is not true by finding a counterexample. “The sum of any 3 consecutive integers is even" 5. Show a Proof by exhaustion for the following: For n = 2, 4, 6, n²-1 is odd 6. Show an informal Direct Proof for “The sum of 2 even integers is even.” Recursive Definitions 7. The Fibonacci Sequence is...

Prove that if n is an integer and n^2 is even the n is even.

prove that every integer is either even or odd but never both.

If f(n) = 3n+2 and g(n) = n, then Prove that f(n) = O (g(n))

Prove that the following is true for all positive integers n: n is odd if and...

Prove that the following is true for all positive integers n: n is odd if and only if n2 is odd.

Find a system of recurrence relations for the number of n-digit quaternary sequences that contain an even number of 2’s and an odd number of 3’s.

Find a system of recurrence relations for the number of n-digit quaternary sequences that contain an even number of 2’s and an odd number of 3’s. Define the initial conditions for the system. (A quaternary digit is either a 0, 1, 2 or 3)

Write c code to determine if a binary number is even or odd. If it is...

Write c code to determine if a binary number is even or odd. If it is odd, it outputs 1, and if it is even, it outputs 0. It has to be less than 12 operations. The operations have to be logical, bitwise, and arithmetic.