Set identities. Prove the set identity [18] (A4B)∩C = (A∩C)4(B∩C). Illustrate the identity by drawing Venn’s diagram.

The maximum mark for this coursework is 100. Remember that marks will be awarded for presentation. You must write in clear and concise English sentences, explaining what you are doing. It is not sufficient to merely state answers.
1. Set identities. Prove the set identity [18] (A4B)∩C = (A∩C)4(B∩C). Illustrate the identity by drawing Venn’s diagram. [5]
2. Countable sets. Here you can use the fact that the set Q of rational numbers is countable (you do not need to copy the proof that Q is countable from your lecture notes).
(a) Prove that the Cartesian product of two countable sets is countable. [5]
(b) Prove that the set of points (p,q) on the plane with rational coordinates p and q is countable. [5]
(c) Prove that any infinite set of nonintersecting discs on the plane is countable. [15]
3. Relations. For each of the following relations say whether the relation is reflexive, sym- metric, transitive on the given set. In each case you must explain why your stated answer is correct.
(a) On Z, x ∼ y if 3x + 5y is divisible by 2. [6] (b) On Z, x ∼ y if 3x + 5y is divisible by 3. [6] (c) On Z, x ∼ y if 3x + 5y is divisible by 8. [6] (d) On R, x ∼ y if 3x + 5y is not rational. [6] 4. Functions. Suppose f : A→B and g : B→C are functions. (a) Prove that if g◦f is injective then f is injective. [7] (b) Prove that if g◦f is surjective then g is surjective. [7] (c) Prove that if f is surjective and g is not injective then g◦f is not injective. [7] (d) Prove that if f is not surjective and g is injective then g◦f is not surjective.

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