1.6 Simple deductive proof (IB Maths: Analysis and Approaches HL): Revision notes

Section 1

What is a deductive proof?

A deductive proof starts from facts you already accept (definitions, algebraic rules, earlier results) and uses a chain of logical steps to reach a conclusion that must be true. Checking a few examples is not a proof: $n^2+n+41$ is prime for $n = 0, 1, \ldots, 39$ but not for $n = 40$ .

At SL you need two kinds:

Algebraic proofs, where you show two expressions are always equal.
Numerical proofs about integers, for example that the sum of three consecutive integers is always a multiple of 3.

Key termsdeductive proof

Common mistake

Testing $n=1, 2, 3$ and writing “so it is always true”. Examples can support a conjecture or disprove it, but never prove it for all cases.

Section 2

Equality (=) and identity (≡)

An equation such as $x^2 = 4x$ is true only for particular values ( $x = 0$ or $x = 4$ ). An identity is true for every value of the variable, and is written with the symbol $\equiv$ : $(x-3)^2 + 5 \equiv x^2 - 6x + 14.$ When a question says “show that ... $\equiv$ ...” you must prove it holds for all $x$ , not solve it.

Key termsidentityequation

Exam tip

If substituting two or three values gives the same result on both sides, the identity is plausible — now prove it algebraically.

Section 3

Laying out an LHS to RHS proof

Start with one side (usually the more complicated one), transform it using known algebraic steps, and finish with the other side:

LHS $= (x-3)^2 + 5$ $= x^2 - 6x + 9 + 5$ $= x^2 - 6x + 14 =$ RHS.

Each line follows from the one before. Do not write the whole identity on every line and operate on both sides — that assumes what you are trying to prove.

Key termsLHSRHS

Common mistake

Starting with $(x-3)^2+5 = x^2-6x+14$ and simplifying both sides until you get $0 = 0$ . This is not an acceptable LHS to RHS proof.

Example

Show $(2x-5)^2 - 4(x-1)(x-4) \equiv 9$ : LHS $= 4x^2-20x+25 - 4(x^2-5x+4) = 4x^2-20x+25-4x^2+20x-16 = 9 =$ RHS.

Section 4

Proofs about integers

Write general integers algebraically:

any integer: $n$ ; consecutive integers: $n$ , $n+1$ , $n+2$ (or $n-1$ , $n$ , $n+1$ )
an even number: $2n$ ; an odd number: $2n+1$ ; consecutive odd numbers: $2n+1$ , $2n+3$
a multiple of $k$ : $k\times$ (an integer)

Then simplify and factorise to show the required property. For example $(2m+3)^2-(2m+1)^2 = 8m+8 = 8(m+1)$ , and because $m+1$ is an integer, this is a multiple of 8. Always state why the bracket is an integer.

Key termsevenoddmultiple

Common mistake

Using $2n+1$ and $2n+3$ for two unrelated odd numbers — that forces them to be consecutive. Use different letters ( $2a+1$ , $2b+1$ ) when the numbers are independent.

Section 5

Checking a result

The syllabus expects you to check results. After proving an identity, substitute a convenient value and confirm both sides agree: for $P + 1 \equiv (2m+2)^2$ with $P=(2m+1)(2m+3)$ and $m = 4$ , $9\times 11 + 1 = 100 = 10^2$ .

A check does not replace the proof, but it catches expansion and sign errors. Choose values that are easy to work with but not special cases like $0$ or $1$ , which can hide mistakes.

Key termscheck

Exam tip

In a number-trick question, run the trick with one actual number at the end to confirm your algebra.

Must know

An example is not a proof; a deductive proof covers every case.
$=$ may hold only for some values; $\equiv$ holds for all values.
LHS to RHS: begin with one side, finish with the other, one valid step per line.
Integers: $n$ , $n+1$ ; even $2n$ ; odd $2n+1$ . Factorise to show a multiple, and say why the factor is an integer.
Check your result with a numerical substitution.

That's the notes covered.

Carry on to the next subtopic.