The document discusses different methods of proof in mathematics including direct proof, proof by contradiction, and proof of the contrapositive. It provides examples of each. Direct proof involves assuming a statement P is true and using logical steps to prove another statement Q is true. Indirect proof involves proving the contrapositive or using proof by contradiction. The contrapositive of "if P then Q" is "if not Q then not P". Proof by contradiction assumes the opposite of the statement to be proven and shows it leads to a contradiction. The document also includes exercises asking students to match logical equivalences and complete proofs using logical steps.

1. Writing Proofs
General Mathematics

2. Lesson Objectives
At the end of the lesson, the students must be
able to:
• illustrate the different methods of proof and
disproof; and
• justify mathematical and real-life statements
using the different methods of proof and
disproof.

3. Direct Proof
Suppose you want to prove that P Q. You need
to assume P and then use the rules of inference
axioms, definitions, and logical equivalences
to prove Q.

4. Sample of Direct Proof
Assume that a and b are odd integers. Then a
and b can be written as
a = 2c + 1 and b = 2d + 1 where c and d are
also integers.
Because a + b is twice another integer (c + d +
1), a + b is an even integer
a + b = 2c + 1 + 2d + 1 Substitution
= 2c + 2d + 2 Associative and Commutative
Properties
= 2(c + d + 1) Distributive Property

5. Indirect Proof
The two methods of indirect proof are proof of
the contrapositive and proof by contradiction.
Proof of the contrapositive means that you have
to prove the contrapositive of the given
conditional propositions.

6. Sample of Indirect Proof
Proof: Prove the contrapositive of the
statement:If a and b are both odd
integers, then ab is odd.
Proof: Suppose a and b are odd integers, then a
= 2c + 1 and b = 2d + 1, where c and d are
integers and

7. Sample of Indirect Proof
ab = (2c + 1)(2d + 1)
= 4cd + 2c + 2d + 1
= 2(2cd + c + d) + 1
Because ab is twice an integer (2cd + c + d) plus
1, ab is odd.
Proof by contradiction will be discussed in latter
parts of this lesson

8. Exercise A
A B
1. P ∨ T ≡ T a. Commutative Law
2. P ∨ P ≡ P b. De Morgan's Law
3. P ∨ Q ≡ Q ∨ P c. Implication Law
4. ~P ∨ ~Q ≡ ~(P ∧ Q) d. Negation Law
5. (P Q) ≡ ~P ∨ Q e. Domination Law
6. ~P ∧ P ≡ F f. Identity Law
7. (P ∧ Q) ∨ F ≡ P ∧ Q g. Idempotent Law
Match Column A to Column B.

9. Exercise B
Complete the following and explain the logical
equivalences used.
1. a. A ~B
b. C D
c. B ∧ A /∴ ~C
2. a. (A ~B) ∧ (B C)
b. C A
c. ~D B /∴ D