Advice on Proofs
Proofs Workshop
Finding a method of proof
(a) Read the question carefully: Make sure you know exactly what it is you are being asked to prove. (b) Shape: Try to identify the 'shape' of the proposition, perhaps in terms of formal logic notation. It will help to crystallise your ideas, and may suggest a proof strategy. (c) Examples: Do you understand the proposition? Try a few examples to find out when the result holds and when it does not. (d) Analogies: Is there a similar or analogous theorem whose proof method you can adapt? (e) Pictures: Draw a diagram if it helps. (f) Specialise: Try specialising to a particular case which can be more fully understood (eg: put in some numbers), then broaden to the more general situation. (g) Cases: Consider separate cases (eg: x < 0, x = 0, x > 0). (h) What if: If you are having difficulty proving something, see what happens if it is false. It might indicate a path through the difficulty. (i) Skin a cat: Remember that there may be several different ways of proving a proposition. Unless the question asks for a specific method, it does not matter which method you use - as long as your argument is sound.
Presenting proofs in exams and assignments
(a) Structure: Try to make the structure of your proof clear in your argument, so that the reader is in no doubt about what you are proving and how you are proving it. (b) Labels: Labelling various objects (propositions, sets, functions) may help to structure your argument. Make sure you clearly define any labels you use. (c) Method: If you are asked to use a particular method of proof, you will get no marks if you use a different method, correct or not. (d) Theorems: You may quote, without proof, any theorems or results that have appeared in previous units. Always give a reference and make sure you comply with (e) below. (e) Conditions: Show explicitly that the proposition you are trying to prove satisfies all the conditions required for any theorem you have quoted. (f) Justify: Justify key points of your proof. How much detail to give is a matter of judgement, e.g. is it necessary to justify the statement
Strategies:
p⇒ q
Notation:
(x − a )2 ≥ 0 ?
(Probably not).
Letters p, q, r are used to symbolise propositions or properties. P is a proposition, and P(x) means x satisfies a property p. The symbol ¬ is negation; so ¬p means not-p, or p is false. Other notation is explained in the text as it occurs.
if p, then q,
p implies q,
p only if q,
q if p, whenever p is true, q must true
a) Assume p is true and deduce that q must be true or b) Prove the contrapositive: ¬q ⇒ ¬p (assume q is false, deduce p must be false) (the truth values for p ⇒ q and ¬q ⇒ ¬p are the same) or c) Try to think of an intermediate step r:
( p ⇒ r ) and (r ⇒ q) logically implies p ⇒ q . This may mean either working 'forwards' from p (find r such that p ⇒ r ) or working 'backwards' from q (find r such that r⇒q) p⇔ q
p if and only if q, p iff q a) Prove p ⇒ q and q ⇒ p or b) It may be easier to prove one of these by the contrapositive - ie prove (p ⇒ q and ¬p ⇒ ¬q ) or prove ( q ⇒ p and ¬q ⇒ ¬p ) or c) It may be possible to show directly that either both are true or neither is true.
Shrleen Stibbe
http://shirleenstibbe.co.uk
p
Prove by contradiction: assume ¬p (not-p, ie p is false) and deduce a contradiction.
p∧ q
p and q, p and q are both true Prove p is true and q is true independently
p∨q
p or q (or both - ie at least one of p and q is true) a) Consider cases or b) Prove ¬p ⇒ q (assume p is false and deduce q must be true), or prove ¬q ⇒ p (assume q is false and deduce p must be true)
∀x P ( x )
For all (every) x, P(x) is true a) To prove: prove P(a) is true for all (or an arbitrary) a. b) To disprove: find a particular a for which P(a) is false [ie prove ∃x ¬P(x) , proof by counter-example]
∃x P ( x )
There exists (at least one) x such that P(x) is true a) To prove: prove P(a) for a particular a. b) To disprove: prove P(x) is false for all x [ie prove ∀x ¬P(x) ]
∃1 x P( x )
There exists exactly one (unique) x such that P(x) is true
or ∃ ! x P ( x )
a) Prove ∃x P(x) and b) ∀x ∀y (P( x) ∧ P( y)) ⇒ ( x = y) ie if both P(x) and P(y) are true, then x = y.
A⊆ B
A is a subset of B Show that every a ∈ A is also in B [ie ∀a : (a ∈ A ⇒ a ∈ B) ]
A = B, where A and B are sets
a) Show A ⊆ B and B ⊆ A [ie ∀a : (a ∈ A ⇒ a ∈ B) and ∀b : (b ∈ B ⇒ b ∈ A)] or b) Show A ⊆ B and |A| = |B| (Only possible for finite sets) NB: it is not enough just to show ∀a : (a ∈ A ⇒ a ∈ B) (ie A ⊆ B); you must close the deal.
Induction
To prove P(n) is true for all n ≥ a, where n and a are integers: a) Prove P(a) is true, and b) Weak form: Assume P(k) is true for some integer k ≥ a. Deduce that P(k+1) is true or Strong form: Assume P(r) is true for all integers r, where a ≤ r ≤ k for some k ≥ a. Deduce that P(k+1) is true.
Shirleen Stibbe
http://shirleenstibbe.co.uk