Question

Proof of If and Only if (IFF) and Contrapositive

Let x,y be integers. Prove that the product xy is odd if and only if x and y are both odd integers.

Proof by Contradiction

Use proof by contradiction to show that the difference of any irrational number and any rational number is irrational. In other words, prove that if a is irrational and b is a rational numbers, then a−b is irrational.

Direct Proof

Using a direct proof, prove that:

The sum of two consecutive perfect squares is odd.

***Consider the set of integer numbers.***

***A perfect square is a number that can be expressed as the product of two equal integers.***

Answer 1

1)Proof of If and Only if (IFF) and Contrapositive

Let x,y be integers. Prove that the product xy is odd if and only if x and y are both odd integers.

Proof:

Use Proof by Contrapositive ,

Suppose x and y are both not odd integers.That is,they are even integers

If x=2a and y=2b for some integers a and b

Thus x.y=(2a).(2b)=4ab=2(2ab) for some integer 2ab

Therefore xy is even

That is xy is not odd.

By contrapositive,We proved that  product xy is odd if and only if x and y are both odd integers

by proving product xy is even if and only if x and y are both even integers

2)Proof by Contradiction

Use proof by contradiction to show that the difference of any irrational number and any rational number is irrational. In other words, prove that if a is irrational and b is a rational numbers, then a−b is irrational.

Proof:Use Proof by contradiction.

Suppose the statement is false.

That is, suppose there is a rational number b and an irrational number a such that ,a-b, is rational.

By the definition of rational numbers, we have b = x/y and a-b = p/q for some integers x,y,p,q, where y≠0 and q≠0. That is, a-x/y= p/q.

Then, by basic algebra a=(py+qx)/qy . Because the set of all integers are closed under addition, subtraction, and multiplication, and x,y,p,q are integers, py+qx and qy are also integers. Furthermore, because y≠0 and q≠0, qy≠0. Therefore, by the definition of rational numbers, a is a rational number. On the other hand, in the supposition at the beginning, we supposed a is an irrational number.

Thus, a is a irrational number and b is an rational number, which is a contradiction. So, the supposition cannot be true.

That is, the original statement is true.

3)  Direct Proof

Using a direct proof, prove that:

The sum of two consecutive perfect squares is odd.

Proof:Since n and m are consecutive squares ,then n=a² and m=b² for some integers a,b by a definition of square

Then n+m=a²+b²

=2k+1,where k∈Z⁺

For Example, 25=5² and 36=6²

25+36=61=2(30)+1

Therefore, The given statement "The sum of two consecutive perfect squares is odd" is true