1.2 Proof by deduction — Learn

Prerequisites: expanding brackets; factorising; collecting like terms; substituting values into an expression; the structure of an implication from 1.1.

By the end of this section you can: prove a general statement by working from the assumption with algebra; choose a general letter for a number of a given type; and write a conclusion that links the algebra back to the statement.

1. What deduction is

Proof by deduction starts from the assumption and moves forward one valid step at a time until the conclusion is reached. Each step must follow from the step before.

The key move is to work with a general representative. You do not test a few numbers. You choose a letter for an unknown number and use facts that hold for every value of that letter.

The table is the whole tool kit. An even number is a multiple of $2$, so it can always be written as $2n$. An odd number is one more than an even number, so it can always be written as $2n + 1$.

2. A first deduction

Example 1. Prove that the sum of any two odd numbers is even.

Let the two odd numbers be $2m + 1$ and $2n + 1$, where $m$ and $n$ are whole numbers.

Their sum is

Remove the brackets by collecting like terms:

Take out the common factor $2$:

The number $m + n + 1$ is a whole number, because $m$ and $n$ are. So the sum is a multiple of $2$, and therefore even. This holds for every pair of odd numbers, because $m$ and $n$ were never given particular values.

3. Using the assumption

A deduction starts from the assumption, so the first line is fixed for you. Read the assumption, choose the general form that matches it, and write it down.

Example 2. Prove that $n^2 + n$ is even for every whole number $n$.

Factorise the expression:

The numbers $n$ and $n + 1$ are consecutive, so one of them is even. A product with an even factor is even, so $n(n + 1)$ is even, and $n^2 + n$ is even.

4. Clearing the algebra out of the way

Sometimes the statement is about the value of an expression for every $x$. Then the deduction often starts by rewriting the expression into a shape whose sign you can read.

Example 3. Prove that $x^2 - 4x + 7$ is positive for every real number $x$.

Complete the square:

A square is never negative, so

Adding $3$ gives

Since $3 > 0$, the value of $x^2 - 4x + 7$ is at least $3$, so it is positive for every $x$.

Example 4. Prove that $(n + 1)^2 - n^2 = 2n + 1$ for every whole number $n$.

Expand the bracket:

The $n^2$ terms cancel:

So the difference is $2n + 1$, exactly as claimed.

Watch out. Two traps sit in this section.

Testing a value instead of proving. Writing "try $n = 3$ and it works" proves nothing about the other values of $n$. Use the general letter, and keep it instead of a number.

Starting from the thing you are asked to prove. If you begin with $(n + 1)^2 - n^2 = 2n + 1$, you have assumed the result. Begin with the expression on its own, then simplify it.

Re-check: does every line follow from the one above, and does the last line name the conclusion in words?

Progress check

Claim A in the chapter problem says that $n^2 + n$ is even for every whole number $n$.

1. What general form does an even number take? What general form does a whole number take?
2. Explain why $n(n + 1)$ must contain an even factor.
3. Write the final sentence of the proof in your own words.

You should now be able to

• start a proof from the assumption, using a general letter;
• write an even number as $2n$ and an odd number as $2n + 1$;
• chain algebra steps, with a reason for each one;
• complete the square to prove that an expression is always positive;
• finish with a sentence that links the algebra to the original statement.