# 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.
Step-1 Show that the statement p is true.
Step-2 Show that the statement 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.
Case 1 By assuming that p is false, show that q must be true.
Case 2 By assuming that 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.
Case 1 By assuming that p is true, prove that q must be true. (Direct method)
Case 2 By assuming that 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-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