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.
Theory of Knowledge
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 $m ≠ 0, -1$
Note
- 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.
Exam technique
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.
Example
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, $(x-3)^2 + 5 ≡ x^2 - 6x + 14$
Hint
- 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.
Symbols and Notation for Equality and Identity
Understanding the correct use of mathematical symbols is crucial for clear and precise communication in proofs.
Equality (=)
- The equality symbol (=) is used to show that two expressions have the same value.
- This may be true for all values of a variable or only for specific values.
Example
- $x^2 = 4$ is true when $x = 2$ or $x = -2$, but not for all values of $x$.
- For example when $x=3$ then $x^2\not = 4$
Identity (≡)
The identity symbol (≡) is used to show that two expressions are equivalent for all possible values of the variables (except where undefined).
Example
$(a+b)^2 ≡ a^2 + 2ab + b^2$ is true for all real values of $a$ and $b$.
Common Mistake
- Students often confuse equality (=) with identity (≡).
- Remember that an identity is always true for all possible values, while an equality may only be true for specific values.
Checking Results
- An important aspect of mathematical proof is the ability to verify results, including one's own work.
- This involves several strategies:
- Substitution: For algebraic proofs, substitute various values for the variables to check if the equation holds true.
- Reverse engineering: Work backwards from the result to see if you arrive at the starting point.
- Alternative methods: Solve the problem using a different approach and compare the results.
- Graphical verification: For equations, graph both sides to see if they produce the same curve.
Note
- While checking specific values can increase confidence in a proof, it does not constitute a complete proof.
- A formal algebraic proof is still necessary to establish the general truth of an identity.
