Simple Deductive Proof
- Simple deductive proof is a fundamental concept in mathematics that involves demonstrating the truth of a statement through logical reasoning.
- At SL level, proofs will be limited to showing that one side of an equation is equal to another.
In mathematics, new knowledge can only be derived from preexisting knowledge through proofs. What does this mean for the validity of mathematical knowledge?
Numerical Proofs
- Numerical proofs involve demonstrating the truth of a mathematical statement using specific numbers.
- These proofs are often used to verify simple arithmetic relationships.
Example question
Prove that $\frac{1}{4} + \frac{1}{12} = \frac{1}{3}$.
Solution
- Start with the left-hand side: $\frac{1}{4} + \frac{1}{12}$
- Find a common denominator: $\frac{3}{12} + \frac{1}{12}$
- Add the fractions: $\frac{4}{12}$
- Simplify: $\frac{1}{3}$
- Thus, we have shown that $\frac{1}{4} + \frac{1}{12} = \frac{1}{3}$
Algebraic Proofs
- Algebraic proofs involve demonstrating the truth of a mathematical statement using variables and algebraic manipulations.
- These proofs are more general and apply to a wider range of cases.
Example question
Prove that $\frac{1}{m+1} + \frac{1}{m^2+m} ≡ \frac{1}{m}$ for all $m ≠ 0, -1$.
Solution
- Start with the left-hand side: $\frac{1}{m+1} + \frac{1}{m^2+m}$
- Find a common denominator: $\frac{m^2+m}{(m+1)(m^2+m)} + \frac{m+1}{(m+1)(m^2+m)}$
- Add the fractions: $\frac{m^2+2m+1}{(m+1)(m^2+m)}$
- Factor the numerator: $\frac{(m+1)^2}{(m+1)(m^2+m)}$
- Cancel $(m+1)$ from numerator and denominator: $\frac{m+1}{(m^2+m)}$
- Factor out $m$ from the denominator: $\frac{m+1}{m(m+1)}$
- Cancel $(m+1)$ from numerator and denominator: $\frac{1}{m}$
- Thus, we have shown that $\frac{1}{m+1} + \frac{1}{m^2+m} ≡ \frac{1}{m}$ for all $m ≠ 0, -1$
- The symbol ≡ is used to denote an identity, which means the equation is true for all values of the variable (except where undefined).
- This is different from the equality symbol =, which may only be true for specific values.
Left-Hand Side to Right-Hand Side (LHS to RHS) Proofs
- LHS to RHS proofs are a structured way of presenting algebraic proofs.
- The goal is to start with the expression on the left-hand side of the equation and transform it step-by-step until it matches the right-hand side.
Steps for LHS to RHS Proofs
- Begin with the left-hand side expression.
- Apply valid algebraic manipulations (e.g., factoring, expanding, simplifying).
- Continue transforming the expression until it matches the right-hand side.
- Ensure each step is justified and clearly explained.
To prove that $(x-3)^2 + 5 ≡ x^2 - 6x + 14$:
$$LHS = (x-3)^2 + 5 = (x^2 - 6x + 9) + 5 \text{ (expanding the square)}$$
$$= x^2 - 6x + 14 \text{ (combining like terms)}$$
$$= RHS$$
Therefore, $(x-3)^2 + 5 ≡ x^2 - 6x + 14$
Tip- When performing LHS to RHS proofs, it's often helpful to work from both sides simultaneously, meeting in the middle.
- This can make the proof more efficient and easier to follow.
