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2q C 1/ for some integer q, and hence, x y is an odd integer. Consequently, it has been proven that if x and y are odd integers, then x y is an odd integer. Writing Guidelines for Mathematics Proofs At the risk of oversimplification, doing mathematics can be considered to have two distinct stages. The first stage is to convince yourself that you have solved the problem or proved a conjecture. This stage is a creative one and is quite often how mathematics is actually done. The second equally important stage is to convince other people that you have solved the problem or proved the conjecture.

P is necessary and sufficient for Q. 40 Chapter 2. Logical Reasoning Tautologies and Contradictions Definition. A tautology is a compound statement S that is true for all possible combinations of truth values of the component statements that are part of S . A contradiction is a compound statement that is false for all possible combinations of truth values of the component statements that are part of S . That is, a tautology is necessarily true in all circumstances, and a contradiction is necessarily false in all circumstances.

P ^ Q/ and :P _ :Q logically equivalent? 3. Suppose that the statement “I will play golf and I will mow the lawn” is false. Then its negation is true. Write the negation of this statement in the form of a disjunction. Does this make sense? Sometimes we actually use logical reasoning in our everyday living! Perhaps you can imagine a parent making the following two statements. 44 Chapter 2. Logical Reasoning Statement 1 Statement 2 If you do not clean your room, then you cannot watch TV. You clean your room or you cannot watch TV.