WJEC A-Level Mathematics

## What a proof is A **proof** is a chain of logical steps that starts from given assumptions, or from results already established, and ends at the statement to be shown. Each step must follow from the one before, and the conclusion has to hold in every case covered, not just in the cases tried. A few examples that work prove nothing on their own; a single example that fails is enough to disprove a general claim. Three methods are used at this level. ### Proof by deduction Start from known facts and use algebra to reach the result. To prove something about a family of integers, represent them algebraically: an even number is $2n$, an odd number is $2n+1$, and consecutive integers are $n$ and $n+1$, where $n$ is an integer. Claim: the sum of two consecutive odd numbers is a multiple of 4. $$ \begin{aligned} (2n+1) + (2n+3) &= 4n + 4 \\ &= 4(n+1) \end{aligned} $$ Since $n+1$ is an integer, the sum is a multiple of 4. Claim: the square of any odd number is one more than a multiple of 8. Write the odd number as $2n+1$. Then $(2n+1)^2 = 4n^2 + 4n + 1 = 4n(n+1) + 1$. One of $n$ and $n+1$ is even, so $n(n+1)$ is even and $4n(n+1)$ is a multiple of 8. The laws of logarithms are also proved by deduction, from the definition that $x = a^n$ means $n = \log_a x$ (for $a > 0$, $x > 0$). Let $x = a^m$ and $y = a^n$, so that $\log_a x = m$ and $\log_a y = n$. * The product: $xy = a^m \times a^n = a^{m+n}$, so $\log_a(xy) = m + n = \log_a x + \log_a y$. * The quotient: $\dfrac{x}{y} =…