SL 1.6 — Deductive Proof & Mathematical Identity
| Focus | What It Means |
|---|---|
| Deductive Proof | A logical step-by-step argument showing that a statement must be true. |
| LHS → RHS Proof | Start from one side of an equation and transform it into the other using algebra. |
| Identity vs Equality | An identity is always true; an equality may only be true for some values. |
📌 What Is a Deductive Proof?
A deductive proof is a logically ordered sequence of algebraic steps that shows why a mathematical statement must be true.
Each step must follow directly from a known rule such as simplification, factorisation, expanding brackets, or cancelling terms.
- It does not test with numbers — it proves generally.
- Every step must be mathematically justified.
- Shortcuts or guessing invalidate the proof.
Logical proof is used in computer programming, cryptography, legal argumentation, and artificial intelligence validation.
📌 LHS → RHS Proof (Left-Hand Side to Right-Hand Side)
In this method, you start only with the left-hand side of the identity and apply algebraic rules until it becomes identical to the right-hand side.
You never assume the RHS is true during the process.
Numerical Example:
1 ÷ 4 + 1 ÷ 12
= 3 ÷ 12 + 1 ÷ 12
= 4 ÷ 12
= 1 ÷ 3
Algebraic Generalisation:
1 ÷ (m + 1) + 1 ÷ (m² + m)
= (m + m + 1) ÷ [m(m + 1)]
= 1 ÷ m
Never write LHS = RHS at the start.
You must demonstrate how LHS becomes RHS through valid algebraic steps.
📌 Algebraic Identity Proof
Example:
(x − 3)² + 5
= x² − 6x + 9 + 5
= x² − 6x + 14
Since both sides match exactly for all values of x, this is an identity.
- An identity is true for every possible value.
- An equation is only true for specific solutions.
Is mathematical proof more certain than scientific proof, given that no physical experiment is required?
📌 Notation: Equality vs Identity
- = means the two expressions are equal for a particular value.
- ≡ means the two expressions are equal for all values.
Example:
(x − 3)² + 5 ≡ x² − 6x + 14
Since it is always true, not just for some x.
If the question says “Show that”,
you must use structured reasoning — numerical checking alone earns zero credit.
Proof methods can support algebraic modelling investigations by guaranteeing that transformations preserve correctness.