Edexcel IAL - Pure Maths 2- 1.2 Proof by Exhaustion- Study notes - New syllabus
Edexcel IAL – Pure Maths 2- 1.2 Proof by Exhaustion -Study notes- New syllabus
Edexcel IAL – Pure Maths 2- 1.2 Proof by Exhaustion -Study notes -Edexcel A level Physics – per latest Syllabus.
Key Concepts:
- 1.2 Proof by Exhaustion
Proof by Exhaustion
Proof by exhaustion is a method of proof used when the number of possible cases is finite and reasonably small. The statement is proven by checking every possible case and showing that the result holds in each one.
This method is especially useful when variables are restricted to a small set of values.
When to Use Proof by Exhaustion
| Situation | Reason |
| Finite number of cases | All possibilities can be listed and checked. |
| Small integer sets | Such as odd numbers less than 7, digits 1–9, etc. |
| Direct proof not required | Checking all cases is simpler than forming a general algebraic argument. |
Structure of a Proof by Exhaustion
- State clearly all possible cases.
- Check the result for each case.
- Show that the statement holds in every case.
- Conclude that the statement is true.
Example
Suppose \( x \) is an integer such that \( 1 \le x \le 4 \). Prove that \( x^2 + x \) is even.
▶️ Answer / Explanation
Possible values of \( x \):
\( 1,\ 2,\ 3,\ 4 \)
Check each case:
- \( x = 1 \Rightarrow x^2 + x = 2 \)
- \( x = 2 \Rightarrow x^2 + x = 6 \)
- \( x = 3 \Rightarrow x^2 + x = 12 \)
- \( x = 4 \Rightarrow x^2 + x = 20 \)
Each result is even.
Conclusion: For all integers \( x \) with \( 1 \le x \le 4 \), \( x^2 + x \) is even.
Example
Suppose \( n \) is an integer such that \( -2 \le n \le 2 \). Prove that \( n^2 \le 4 \).
▶️ Answer / Explanation
Possible values of \( n \):
\( -2,\ -1,\ 0,\ 1,\ 2 \)
Check each case:
- \( (-2)^2 = 4 \)
- \( (-1)^2 = 1 \)
- \( 0^2 = 0 \)
- \( 1^2 = 1 \)
- \( 2^2 = 4 \)
In all cases, \( n^2 \le 4 \).
Conclusion: For all integers \( n \) with \( -2 \le n \le 2 \), the inequality holds.
Example
Suppose \( x \) and \( y \) are odd integers less than 7. Prove that \( x^2 + y^2 \) is divisible by 2.
▶️ Answer / Explanation
Odd integers less than 7:
\( 1,\ 3,\ 5 \)
Possible squares:
- \( 1^2 = 1 \)
- \( 3^2 = 9 \)
- \( 5^2 = 25 \)
Now check all possible sums:
- \( 1 + 1 = 2 \)
- \( 1 + 9 = 10 \)
- \( 1 + 25 = 26 \)
- \( 9 + 9 = 18 \)
- \( 9 + 25 = 34 \)
- \( 25 + 25 = 50 \)
Each sum is divisible by 2.
Conclusion: The expression \( x^2 + y^2 \) is divisible by 2 for all odd integers \( x \) and \( y \) less than 7.