Proof Cheat Sheet - Edexcel Maths A-level - Year 1 Pure

Introduction to Mathematical Proofs

• Proof: A logical argument showing that a statement is always true.
  • Theorem: A proven statement.
  • Conjecture: A statement yet to be proven.
• Key steps in constructing a proof:
  • State any information or assumptions.
  • Show every step clearly.
  • Follow logical progression from one step to the next.
  • Cover all possible cases.
  • State a conclusion.

Types of Proofs

Proof by Deduction

• Start from known facts and use logical steps to conclude.
• Example 1: Prove (x^2) is even for all x.
  • Assume x is odd or even and show it leads to (x^2) being even.
• Example 2: Prove (ax^2 + bx + c = 0) has no real roots under certain conditions.
  • Use inequalities and graph sketching to show no real roots exist.

Proof by Identity

• Match one side of an equation to the other using algebraic manipulation.
• Example 3: Prove a polynomial identity by expanding and simplifying both sides to show they are equal.

Proof by Exhaustion

• Break statement into cases and prove each separately.
• Example 5: Prove square numbers are either multiples of 4 or one more.
  • Consider odd and even cases separately and show each result.

Counter-example

• Show a statement is false using a single contradicting example.
• Example 6: Disprove an inequality by choosing negative values.
• Example 7: Disprove a statement about prime numbers by showing 2+3 is odd.

Conclusion

• Mathematical proofs require a systematic approach to show truth or falsehood.
• Various methods are used based on the nature of the statement.

• Additional resources and references:
  • Physics and Maths Tutor
  • Cheat Sheets
  • Proof Cheat Sheet Document