The document outlines the lesson objectives for teaching different methods of mathematical proofs, including direct proof and indirect proof techniques like proof of the contrapositive and proof by contradiction. It provides examples illustrating how to apply these methods to prove statements about integers, particularly addressing odd and even integers. Additionally, it contains exercises aimed at reinforcing the concepts and logical equivalences related to mathematical proofs.

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
∴