Use proof by contrapositive to prove the following: a. For every integer n, if n

ID: 3120086 • Letter: U

Question

Use proof by contrapositive to prove the following:

a. For every integer n, if n^2 is an odd, then n is odd.

b. If x and y are real numbers and x + y is irrational, then x is irrational or y is irrational.

c. If x and y are positive real numbers and xy > 400, then x > 20 or y > 20. Exercise 3:

Use proof by contradiction to prove the following:

a. If a group of 9 kids have won a total of 100 trophies, then at least one of the 9 kids has won at least 12 trophies.

b. 32 is irrational. You can use the following fact in your proof: “If n is an integer and n3 is even, then n is even.”

c. There is no smallest integer.

Solution:

(a)

If the square of an integer n is odd, then integer is odd.

thus

for all integers n, if n^2 is odd , then n is odd.

Step 1. Form the contrapositive of the given statement. That is,

For all integers n, if n is not odd, then n^2 is not odd

But, we know that an integer is not odd if, and only if, it is even .

So, the contrapositive becomes

For all integer n, if n is even, then n^2 is even

Step 2. Now prove the contrapositive using method of direct proof:

Suppose n is [particular but arbitrarily chosen] integer. [We must show that n^2 is also odd.]
By definition of even , we have

n = 2k for some integer k.

Then by substitution, we have

n . n = (2k) . (2k)

= 4k^2

= 2.(2k^2)

Now 2(2k^2) is an integer. Hence, we have a form:

n . n = 2 . (some integer)

or n^2 = 2 . (some integer)

and so by definition of even is n^2 even.

Step 3.

Therefore, the given statement is true by the logical equivalence between a statement and its contrapositve.

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