Proof by deduction
In a nutshell
Proof by deduction (or direct proof) involves proving a statement to be true using algebra. There are different expressions that you need to know which help represent numbers in maths.
Expressions for proof
Here are some expressions for numbers that you need to know.
| DESCRIPTION
| EXPRESSION
|
| An even number
| 2n
|
| An odd number
| 2n+1
|
| A multiple of x
| xn
|
| A square number
| n2
|
| Two consecutive numbers
| n,n+1
|
| Two random numbers
| n,m
|
Example 1
Represent two consecutive odd squares algebraically.
Work step by step, going through the description.
Represent two consecutive numbers:
n,n+1
These numbers have to be odd, so double them (making them even), then add 1:
2(n)+1,2(n+1)+1
2n+1,2n+3
These numbers have to be square numbers, so square them:
(2n+1)2,(2n+3)2
Two consecutive odd squares are (2n+1)2 and (2n+3)2.
Algebraic proof
To prove something algebraically, use the expressions to represent each number with algebra. Then, manipulate the expression until it's in the desired form.
Example 2
Prove that the difference between two consecutive odd squares is always a multiple of 8.
Two consecutive odd squares can be represented as:
(2n+1)2,(2n+3)2
As shown in the above example. Their difference can be found by expanding the brackets and simplifying.
(2n+3)2−(2n+1)2=(2n+3)(2n+3)−(2n+1)(2n+1)=(4n2+12n+9)−(4n2+4n+1)=8n+8=8(n+1)
The expression simplifies to 8(n+1), which is a multiple of 8 because it is in the form 8× an integer.
Therefore, the difference between two consecutive squares is always a multiple of eight.
Proof with inequalities
Algebraically manipulating inequalities is the same as algebraically manipulating equations. The important rule to remember is that when multiplying or dividing by a negative number, reverse the inequality symbol.
To prove a statement that involves an inequality, it's important to remember the order of logic.
Note: You may have to use the ⇒ symbol to formally show your reasoning.
Example 3
Prove that 2x+1 is always positive for all positive values of x.
Rewrite the statement using "if _, then _":
"If x is positive, then 2x+1 is positive"
Rewrite it using arrow notation:
x>0⇒2x+1>0
This means that to prove this formally, begin with the left hand side and use algebra to end up at the right hand side:
x>0⇒2x>0⇒2x+1>1>0⇒2x+1>0
x>0⇒2x+1>0
Inequality proofs with square numbers
Sometimes, you may be asked to prove a result that will require you to use a certain property of square numbers:
Square numbers are positive
Example 4
Prove that, for any value of x, x2≥2x−1.
First, do some rough work to see how to begin the proof.
Try to experiment with the expression:
x2≥2x−1⇒x2−2x+1≥0⇒(x−1)(x−1)≥0⇒(x−1)2≥0 (⋆)
This means that the proof uses the property of square numbers being either positive or 0:
x2≥0
Begin the formal proof using the steps used in the rough work, but in the "correct" direction:
For any value of x, (x−1)2≥0 because it is a square.
(x−1)2≥0⇒(x−1)(x−1)≥0⇒x2−2x+1≥0⇒x2≥2x−1
Therefore, for any value of x, x2≥2x−1.
Note: The reason why the working out at (⋆) is not correct is that you should always aim to end your proof with the expression that you are trying to prove.
