Example of a Contrapositive Proof
Suppose x ∈ Z. If 7x+9 is even, then x is odd.
Proof. (Contrapositive) Suppose x is not odd.
Thus x is even, so x = 2a for some integer a.
Then 7x+9 = 7(2a)+9 = 14a+8+1 = 2(7a+4)+1.
Therefore 7x+9 = 2b +1, where b is the integer 7a+4.
Consequently 7x+9 is odd
Therefore 7x+9 is not even
Explanation of the proof:
The given proof uses the method of proof by contrapositive to show that if 7x+9 is even, then x is odd. Let's break down the proof step by step:
1. Suppose x is not odd. This means x is even, so we can write x = 2a for some integer a.
2. Substitute the value of x into the expression 7x+9 to obtain 7(2a)+9 = 14a+8+1.
3. Rearrange the terms to get 14a+8+1 = 2(7a+4)+1.
4. We can see that 7a+4 is an integer, so let's represent it as b. Then the expression becomes 2b+1.
5. Therefore, we have shown that 7x+9 can be written as 2b+1, where b is an integer (specifically, b = 7a+4).
6. From step 5, we can conclude that 7x+9 is odd because it takes the form of 2b+1, where b is an integer.
7. Consequently, if 7x+9 is odd, it means that 7x+9 is not even.
By proving the contrapositive, we have established that if 7x+9 is even, then x is odd.