# Proof by Deduction

‘If today is a weekend, then tomorrow must be a weekday.’

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This statement can either be true or false, which makes it perfect for proof by deduction. You can split this statement into two parts: Today is a weekend (A); tomorrow must be a weekday (B). Mathematically, you can write it as:

$$A \rightarrow B$$, where $$\rightarrow$$ is the symbol meaning ‘implies’.

## What is proof by deduction?

In Proof by Deduction, the truth of the statement is based on the truth of each part of the statement (A; B) and the strength of the logic connecting each part.

Statement A: ‘if today is a weekend’ gives us two answers, Saturday and Sunday, as these are the only two days of the weekend.

We then use our answers for statement A and statement B to test the logic of the main statement.

If today is Saturday, then tomorrow is a Sunday. Thus, the concluding statement is false. However, if today is Sunday, tomorrow is Monday, and the concluding statement is true.

Therefore, the logic of the concluding statement depends on statement A and is weak as a result.

In Maths, the concluding statements tend to have more conclusive answers (because numbers don’t lie!). To prove a mathematical conclusion (conjecture) by proof of deduction, you need strong mathematical axioms and logic.

Mathematical axioms are the mathematical concepts underlining the concluding statement.

## Solving Proof by Deduction Questions

To solve a Proof by Deduction question, you must:

1. Consider the logic of the conjecture.
2. Express the axiom as a mathematical expression where possible.
3. Solving through to see if the logic applies to the conjecture.
4. Making a concluding statement about the truth of the conjuncture.

### Expressing axiom mathematically

Although most of these algebraic rules will be familiar to you, it is good to stay familiar with them as expressing axioms as a mathematical expression sometimes requires some creativity using these rules.

n stands in for any number.

• To express n is a multiple of A, you can write as An

Express n as a multiple of 12 mathematically.

A is 12. Therefore, the answer is 12n

• To express consecutive numbers, you can start with n (or any other starting point) and add one each time to get n + 1, n + 2, etc.

Express the next two consecutive numbers after $$x^2$$

To get the following consecutive numbers, you add 1 to each consecutive number. Therefore, the first term is $$x^2$$, the second term is $$x^2 + 1$$, the third term is $$x^2 + 2$$.

• To express consecutive even numbers, you can start with the consecutive numbers: n, n + 1, n + 2. You then multiply each term by 2 as all even numbers are multiples of 2. Therefore the consecutive even terms are 2 (n), 2 (n + 1), 2 (n + 2) which can be simplified to 2n, 2n + 2, 2n + 4 etc.
• Expressing consecutive odd numbers is a little bit more complicated than expressing consecutive even numbers as odd numbers are not part of a multiple. However, they are defined by not being a multiple of two; therefore, all the gaps in the consecutive even numbers will make up the consecutive odd numbers.

 Consecutive even numbers 2n 2n + 2 2n + 4 Consecutive odd numbers 2n + 1 2n + 3 2n + 5

## Example of proof by deduction

We will now go through a few examples to show how you answer questions like these.

Prove the sum of two consecutive numbers is equivalent to the difference between two consecutive numbers squared.

As described above, you can algebraically express two consecutive numbers as n, n + 1 .

The sum of two consecutive numbers is therefore $$n + n + 1 = 2n +1$$

To find the difference between two consecutive numbers squared, you first have to square each consecutive number to get $$(n)^2$$ and $$(n + 1)^2$$.

Expanding out and simplifying the squares gives you:

$$(n)^2 \quad becomes \quad n^2$$

$$(n + 1)^2 = (n + 1) (n + 1) = n^2 + 2n + 1$$

Therefore the difference between two consecutive numbers squared is

$$n^2 + 2n + 1 - n^2 = 2n + 1$$

To finish off the question, you must write a concluding statement: The sum of two consecutive numbers and the difference between two consecutive numbers squared is equal to each other as they are both equal to 2n + 1.

Prove the answer to the equation $$x^2 + 8x + 20$$ is always positive.

As you only want one variable of x, you need to complete the square with the equation.

1. First, you halve b (8) and substitute it into your new equation: $$(x + 4)^2$$.
2. You then expand out to find your constant outside the bracket$$(x + 4)^2 = (x + 4)(x + 4) = x^2 + 8x +16$$. You need +20 to make the new equation match the same as the equation, so you need to +4. Therefore, the answer is $$(x + 4)^2 + 4$$

As always, you need a concluding statement to explain the maths: Regardless of the value of x, by squaring it and adding 4, the value of the equation will always be positive.

## Proof by Deduction - Key takeaways

• Proof by deduction uses mathematical axioms and logic to prove or disprove a conjecture.
• You can express several axioms algebraically, like even and odd consecutive numbers.
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How do you use the deduction theorem?

1. Consider the logic of the conjecture.

2. Express the axiom as a mathematical expression where possible.

3. Solve through to see if the logic applies to the conjecture.

4. Make a concluding statement about the truth of the conjecture.

What are the basic rules of deduction?

There are no explicit rules to help prove by deduction. Instead, to prove a conjuncture through deduction, the truth of the second statement is based on the truth of the first.

Who introduced deductive proof?

Deductive reasoning was introduced by Aristotle.

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