## Mathematical Reasoning

# Validating Statements

- Tautology (t) :- The compound statement which contain only T in the last column of their truth table is called tautology, which if denoted by t.

Eg:- (p ∨ q) ∨ (∼p ∨ ∼q) - Contradiction (c):- The compound statement which contain only F in the last column of their truth table is called contradiction which is denoted by c.

Eg: [(p → q) → p] ∧ ∼ p - Rule 1 :
*If p and q are mathematical statements, then in order to show that the statement "p and q" is true, the following steps are followed.**p*is true.*q*is true. - Rule 2 :
*Statements with "Or"*

If*p*and*q*are mathematical statements, then in order to show that the statement "*p*or*q*" is true, one must consider the following.*p*is false, show that*q*must be true.*q*is false, show that*p*must be true. - Rule 3 :
*Statements with "If-then"*

In order to prove the statement "if*p*then*q*" we need to show that*any one*of the following case is true.*p*is true, prove that*q*must be true. (Direct method)*q*is false, prove that*p*must be false. (Contrapositive method) - Rule 4 :
*Statements with "if and only if"*

In order to prove that statement "*p*if and only if*q*", we need to show

(i) If*p*is true, then*q*is true and (ii) if*q*is true, then*p*is true.

### Validating Statements Part-1

### Validating Statements Part-2

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** Algebra of statements**

- commutative laws : p ∨ q ≅ q ∨ p

p ∧ q ≅ q ∧ p - Associative laws : p ∨ (q ∨ r) ≅ (p ∨ q) ∨ r

p ∧ (q ∧ r) ≅ (p ∧ q) ∧ r - Distributive laws : p ∨ (q ∧ r) ≅ (p ∨ q) ∧ (p ∨ r)

p ∧ (q ∨ r) ≅ (p ∧ q) ∨ (p ∧ r) - Demorgan's laws : ∼ (p ∨ q) ≅ ∼p ∧ ∼ q

∼ (p ∧ q) ≅ ∼p ∨ ∼ q

∼ (p ⇒ q) ≅ p ∧ ∼ q

∼ (p ⇔ q) ≅ (p ∧ ∼ q) ∨ (q ∧ ∼ p) - Complement laws : ∼ (∼p) ≅ p

p ∨ (∼p) ≅ t

p ∧ (∼p) ≅ c

∼t ≅ c

∼c ≅ t - Idempotent laws : p ∨ p ≅ p

p ∧ p ≅ p - Identify laws : p ∨ t ≅ t

p ∧ t ≅ p

p ∨ c ≅ p

p ∧ c ≅ c