Question

Please solve the all the questions below. Thanks. Especially pay attention to 2nd question.

t, which type of proof is being used in each case to prove the theorem (A → C)? Last Line 겨 (p A -p) 겨 First Line a C b. C d. (some inference) C Construct a contrapositive proof of the following theorem. Indicate your assumptions and conclusion clearly 2. If you select three balls at random from a bag containing red balls and white balls, then at least two of the selected balls will have the same color 3. Write a direct proof of the following theorem. For all odd integers, n, (nA3 + 3) is even. Let P(x) be the proposition: Given positive integer x, for all positive real numbers a and b, (a + b)1x >= a^x + b^x 4. Prove P(0) using a vacuous proof. 5. Write a proof by contradiction of this theorem: Given positive integers x and y, if (x * y) = 1 then (x = y)

Answer 1

1.

(a) Proof by Contraposition

(b) Propositional Equivalence

(c) Vacuous proof

(d) Proof by rules of inference

2.

P is Select three balls at random from a bag containing red balls and white balls

Q is at least two of the selected balls will have same color

To prove: P -> Q

Contrapositive:

~Q -> ~P

~Q is having a single color.

Ball having single color implies that we can pick two red color balls or two white color balls. So, we are picking two balls from the bag.

~P is picking two balls from the bag which satisfies the condition.

Hence, the given statement is true by contraposition.

3.

Direct proof:

Given that n is odd

• n=2*m+1

n³+3

= (2*m + 1)³ + 3

= 8*m³ + 1 + 6*m + 12*m² + 3

= 8*m³ + 4 + 6*m + 12*m²

= 2* (4*m³ + 2 + 3*m + 6*m²)

Since, n³+3 is a multiple of 2, it must be even.

Hence proved.

4.

X is a positive integer.

Let’s say x is 1.

Left Hand Side = (a+b)^x = (a+b)¹ = a+b

Right Hand Side = a^x + b^x = a¹ + b¹= a+b

Here, Left Hand Side =Right Hand Side

Let’s say x is 2.

Left Hand Side = (a+b)^x = (a+b)² = a²+b²+2*a*b

Right Hand Side = a^x + b^x = a² + b²

Since 2*a*b is an extra in left hand side, always it is greater.

Hence, Left Hand Side>=Right Hand Side

5.

X and y are positive integers.

To prove: if (x*y=1), then x and y are same and are equal to 1.

Proof by Contradiction:

Let’s say x or y are positive integers and one of them atleast is not equal to 1.

So, let’s say x is 1. However, y can’t be 1 as per the condition. So, let’s say y is 2.

Hence, x*y=2 which is not equal to 1.

So, if x*y=1, then x=y and x=y=1.

(Proved)