TMUA logic and proof: a guide with examples

Necessary and sufficient

If 'P implies Q', then P is sufficient for Q and Q is necessary for P. Example: x > 3 is sufficient for x^2 > 9, because any number above 3 has a square above 9. It is not necessary, because x = -4 also gives x^2 = 16 > 9. The condition x^2 > 9 is necessary for x > 3, but not sufficient.

Converse and contrapositive

Take the true statement: if n is divisible by 6, then n is divisible by 3. The converse swaps the parts: if n is divisible by 3, then n is divisible by 6. That is false, as n = 9 shows. The contrapositive negates and swaps: if n is not divisible by 3, then n is not divisible by 6. That is true, and it always has the same truth value as the original. A statement and its converse can differ. A statement and its contrapositive cannot.

Counterexamples

To show a 'for all' statement is false, one example is enough. Claim: for every real x, x^2 > x. Try x = 1/2. Then x^2 = 1/4, which is less than 1/2, so the claim fails. Note that a counterexample cannot prove a statement true, and checking a few examples that work proves nothing.

Quantifiers and negation

The negation of 'for all' is 'there exists', and the negation changes the condition too. The negation of 'every student passed' is 'at least one student did not pass', not 'no student passed'. Likewise, the negation of 'x^2 >= 0 for all real x' is 'there is a real x with x^2 < 0'. Watch the word order in 'for every x there is a y' against 'there is a y for every x'. For example, 'for every integer n there is an integer m with m > n' is true, but 'there is an integer m with m > n for every integer n' is false.

Spotting errors in an argument

Suppose someone solves x^2 = 4x by dividing both sides by x to get x = 4. The step is invalid, because dividing by x is only allowed when x is not zero. The full solution is x(x - 4) = 0, so x = 0 or x = 4, and the argument lost x = 0. When you are asked where an argument goes wrong, check each step for division by something that could be zero, squaring both sides, multiplying by a possibly negative number in an inequality, and assuming what is to be proved.

A method for Paper 2 questions

Our <a href="/tmua/practice-questions/">free practice questions</a> include reasoning examples, and the official past papers give the real style.

• Rewrite the statement in symbols or plain 'if ... then ...' form.
• Decide what each option is claiming: the statement, its converse, its contrapositive or something unrelated.
• Test a claim with a small number before attempting a proof.
• For each wrong option, find the exact reason it fails.