2 TUT2-Proofs Recall some proof techniques: e Toprove " Jdz..... ", say what is. e To prove " %z.... ", begin by fixing a generic =. e Toprove " P =+ ", assume P is true and show that } is true. Write formal, rigorous proofs for these statements: 1. In tutorial 1 , we have defined several concepts. Let A be a non-empty subset of [. We say A is excited if da R such that r 4,z > a. We say A is happy if 20 A such that Yz A,z > a. Below are three claims. Which ones are true and which ones are false? If a claim is true, prove it. If a claim is false, provide a counterexample and a justification of how the counterexample shows the claim is false. {a) If A is happy, then A is excited. (b} If A is excited, then 4 is happy. {c) If A and B are both happy and non-empty with 4 # B, then AU B is happy