1. Methods of Proof Dr. Yasir Ali

2. Proof A (logical) proof of a statement is a finite sequence of statements (called the steps of the proof) leading from a set of premises to the given statement In a mathematical proof, the set of premises may contain any item of previously proved or agreed upon mathematical knowledge (definitions, axioms, theorems, etc.) as well as the specific hypotheses of the statement to be proved. Methods of Proof 1.Direct Proof 2.Method of Contraposition 3.Proof by Contradiction 4.Proof by Counter Example 5.Proof by Cases

3. A direct proof of a statement of the form p → q is a proof that assumes p to be true and then shows that q is true. Proof by contrapositive (indirect proof) of a statement of the form p → q is a proof that assumes that ￢ q is true and then shows that ￢ p is true. That is, a proof of this form is a direct proof of the ￢ q → ￢ p. A proof by contradiction assumes the negation of the statement to be proved and shows that this leads to a contradiction. To disprove a statement of the form “ ∀ x ∈ D, if P(x) then Q(x),” find a value of x in D for which the hypothesis P(x) is true and the conclusion Q(x) is false. Such an x is called a counterexample.

4. An integer n is even if, and only if, n equals twice some integer. An integer n is odd if, and only if, n equals twice some integer plus 1. Symbolically, if n is an integer, then n is even ⇔ ∃ an integer k such that n = 2k. n is odd ⇔ ∃ an integer k such that n = 2k + 1.

5. Proof by contrapositive (indirect proof) of a statement of the form p → q is a proof that assumes that ￢ q is true and then shows that ￢ p is true. That is, a proof of this form is a direct proof of the ￢ q → ￢ p. ￢ q : α is an odd integer (assume true)

6. In a proof by contradiction, the statement ∀ x in D, if P(x) then Q(x) Suppose there is an x in D such that P(x) and ∼ Q(x). Deduce the statement ∼ P(x). But ∼ P(x) is a contradiction to the supposition that P(x) and ∼ Q(x). (Because to contradict a conjunction of two statements, it is only necessary to contradict one of them.)

7. Difference in Proof by Contraposition and Contradiction

10. An integer n is prime if, and only if, n>1 and for all positive integers r and s, if n=rs, then either r or s equals n. An integer n is composite if, and only if, n>1 and n=rs for some integers r and s with 1<r<n and 1<s<n. In symbols: n is prime ⇔ ∀ positive integers r and s, if n = rs then either r = 1 and s = n or r = n and s = 1. n is composite ⇔ ∃ positive integers r and s such that n = rs and 1 < r < n and 1 < s < n

13. Mathematical Induction PRINCIPLE OF MATHEMATICAL INDUCTION To prove that P(n) is true for all positive integers n, where P(n) is a propositional function, we complete two steps: BASIS STEP: We verify that P(1) is true. INDUCTIVE STEP: We show that the conditional statement P(k) → P(k + 1) is true for all positive integers k. This proof technique can be stated as (P (1) ∧ ∀ k(P(k) → P(k + 1))) → ∀ nP (n),