Answer FOUR questions

(b) Let p,q,r be real numbers with q > 0 and let f:\Bbb R®\Bbb R be defined by

2. Let R be an equivalence relation on a set A. Show that the equivalence classes of the relation on A form a partition of A.

Find the fallacy in the following ``proof'' that if R is symmetric and transitive then it is also reflexive:

Let a Î A and let aRb for some b Î A. Then, aRb implies bRa (since R is symmetric). But aRb and bRa together imply that aRa (since R is transitive). Therefore R is reflexive.

Let A = {1,2,3}. Give examples of three relations R₁, R₂, R₃ on A such that

• R₁ is reflexive and symmetric but not transitive;
• R₂ is reflexive and transitive but not symmetric;
• R₃ is symmetric and transitive but not reflexive.

3. (a) Let A Ì B Í \Bbb R, A ¹ B and suppose B is bounded. Explain why the suprema and infima of A and B exist, and show that

(b) Let A,B Í \Bbb R such that a < b for all a Î A, b Î B. Explain why supA and infB exist, and show that

[The Completeness Axiom of Real Numbers may be assumed but, if used, must be clearly stated.]

4. A binary sequence is here defined to be an infinite sequence each of whose terms is either 0 or 1. Let S be the set of all such sequences.

Prove that S is uncountable.

Deduce that

• The set of all subsets of \Bbb N is uncountable.
• The interval [0,1] is uncountable.

5. State and prove the Unique Factorisation Theorem for positive integers.

6. Let f:S® T be a function and let A₁,A₂ Í S and B₁,B₂ Í f(S). Prove, or give a counterexample to disprove, each of the following:

• f(A₁ÈA₂) = f(A₁)Èf(A₂);
• f(A₁ÇA₂) = f(A₁)Çf(A₂);
• f^-1(B₁ÇB₂) = f^-1(B₁)Çf^-1(B₂);
• A₁ = f^-1(f(A₁)).