A-Level Revision - Pure Maths 01
A1 Mathematical Proof - Theory
Mathematical proof is a logical argument demonstrating that a statement is The Specification states students should understand use the structure of
always true. It’s the backbone of mathematical certainty and rigor. mathematical proof, from given assumptions through a series of logical steps
to form a conclusion. Knowledge of a variety of proofs is essential.
There are various ways to prove a mathematical statement, the types of proof are
listed below.
Example Questions - Blind Practice:
1. Direct Proof or Proof by Deduction 1. [Direct Proof] Prove the sum of two different odd numbers is always even
- Direct proof uses known facts, definitions and logical steps to show that a
statement is always true. 2. [Proof by Exhaustion] Prove that for all integers such that ,
- It is commonly used within algebra and number theory. the expression is a prime number.
2. Proof by Exhaustion 3. [Proof by Contradiction] Prove that is irrational
- ‘Trial and Error; method. Involves checking all possible cases individually.
- Only feasible when the number of cases is finite and small. 4. [Proof by Counter Example] ‘ is always multiple of 4’
Use a counterexample to show that the statement is not always true.
3. Proof by Contradiction
- Assume the opposite of what you want to prove.
- Show that this assumption leads to a contradiction, leading to the conclusion
that the original statement must be true.
4. Proof by Counter Example
- Used to disprove a statement
- On evalid counterexample is enough to show a statement is false.
Challenge:
Proof by Induction
- Used to provde statements about integers.
- Involves using a base case, such as proving the statement is true for the first
value such as n=1
- Then using an inductive step, for example assume true for n=k then prove for
n=k+1
, A-Level Revision - Pure Maths 02
A2 Mathematical Proof - Worked Examples
1. Prove that the sum of two different odd numbers is even. 2.Prove for all integers such that , the expression
is a prime number.
* Write out two different odd numbers * * Substitute in all values*
Let 2n+1 and 2m+1 be odd numbers Always provide Prime
a concluding Prime
* Add together the two expressions * statement! Prime This all has to be written!
2n+1+2m+1 = 2n+2m+2
* Factorise out a 2, to write in the form 2k, where k is an integer * For all values of n within the inequality provided, the expression produces
2(n+m+1) = 2k prime outputs, therefore the statement is true.
2k is an even number, therefore the sum of two different odd numbers is even.
3. Prove that is irrational. 4. is always a multiple of 4’
Use a counterexample to show that the statement is not always true.
*Assume the opposite*
*Just simply choose an example that doesn’t work*
Let us assume is rational. Hence can be written as
=6 6 is not a multiple of 4
*Square both sides* So a must be even.
*Concluding statement*
*substitute* , so b must be even
6 is not a multiple of 4, therefore the statement isn’t always true.
must not have been in it’s simplest form, which is a contradiction, so the statement is true.
A1 Mathematical Proof - Theory
Mathematical proof is a logical argument demonstrating that a statement is The Specification states students should understand use the structure of
always true. It’s the backbone of mathematical certainty and rigor. mathematical proof, from given assumptions through a series of logical steps
to form a conclusion. Knowledge of a variety of proofs is essential.
There are various ways to prove a mathematical statement, the types of proof are
listed below.
Example Questions - Blind Practice:
1. Direct Proof or Proof by Deduction 1. [Direct Proof] Prove the sum of two different odd numbers is always even
- Direct proof uses known facts, definitions and logical steps to show that a
statement is always true. 2. [Proof by Exhaustion] Prove that for all integers such that ,
- It is commonly used within algebra and number theory. the expression is a prime number.
2. Proof by Exhaustion 3. [Proof by Contradiction] Prove that is irrational
- ‘Trial and Error; method. Involves checking all possible cases individually.
- Only feasible when the number of cases is finite and small. 4. [Proof by Counter Example] ‘ is always multiple of 4’
Use a counterexample to show that the statement is not always true.
3. Proof by Contradiction
- Assume the opposite of what you want to prove.
- Show that this assumption leads to a contradiction, leading to the conclusion
that the original statement must be true.
4. Proof by Counter Example
- Used to disprove a statement
- On evalid counterexample is enough to show a statement is false.
Challenge:
Proof by Induction
- Used to provde statements about integers.
- Involves using a base case, such as proving the statement is true for the first
value such as n=1
- Then using an inductive step, for example assume true for n=k then prove for
n=k+1
, A-Level Revision - Pure Maths 02
A2 Mathematical Proof - Worked Examples
1. Prove that the sum of two different odd numbers is even. 2.Prove for all integers such that , the expression
is a prime number.
* Write out two different odd numbers * * Substitute in all values*
Let 2n+1 and 2m+1 be odd numbers Always provide Prime
a concluding Prime
* Add together the two expressions * statement! Prime This all has to be written!
2n+1+2m+1 = 2n+2m+2
* Factorise out a 2, to write in the form 2k, where k is an integer * For all values of n within the inequality provided, the expression produces
2(n+m+1) = 2k prime outputs, therefore the statement is true.
2k is an even number, therefore the sum of two different odd numbers is even.
3. Prove that is irrational. 4. is always a multiple of 4’
Use a counterexample to show that the statement is not always true.
*Assume the opposite*
*Just simply choose an example that doesn’t work*
Let us assume is rational. Hence can be written as
=6 6 is not a multiple of 4
*Square both sides* So a must be even.
*Concluding statement*
*substitute* , so b must be even
6 is not a multiple of 4, therefore the statement isn’t always true.
must not have been in it’s simplest form, which is a contradiction, so the statement is true.
