Logic

Logic By Nitin Oke For SAFE HANDS

Nitin Oke Yaa Jatharpeth Road Akola 444005 Ph 0724 2420561

• Subject was developed by George Boole • Logic is also called as “ Boolean Logic” or “Mathematical logic” or “Symbolic logic” • A sentence which is true or false is called statement • If a sentence is true as well as false then is called as “ Paradox” • A single statement is called as “Atomic statement” • When two or more simple statements are connected then are called as compound oe composite statement. Nitin Oke Yaa Jatharpeth Road Akola 444005 Ph 0724 2420561

• p ∧ q (and) Conjunction T ∧ T is T , F otherwise • p ∨ q ( or ) disjunction F ∨ F is F , T otherwise • p ⇒ q implication, T ⇒ F is F otherwise T • P ⇔ q equivalence, biconditional, double implication T ⇔ T , F ⇔ F are T otherwise F • In p ⇒ q means conditional statement – p is sufficient for q – q is necessary for p – P is antecedent or hypothesis or premise – q is consequent or conclusion Nitin Oke Yaa Jatharpeth Road Akola 444005 Ph 0724 2420561

• IF p ⇒ q is given statement then ∼p ⇒ ∼ q is inverse – q ⇒ p is converse ∼q ⇒ ∼ p is contrapositive – q is consequent or conclusion

• p ∧ q (and) Conjunction T ∧ T is T , F otherwise • p ∨ q ( or ) disjunction F ∨ F is F , T otherwise • p ⇒ q implication, T ⇒ F is F otherwise T • P ⇔ q equivalence, biconditional, double implication Nitin Oke Yaa Jatharpeth Road Akola 444005 Ph 0724 2420561

• RULES  p ⇔ q means (p ⇒q) ∧ (q ⇒p)  p ⇒ q means ∼ p ∨ q ∼(p ∧ q) means (∼p) ∨ (∼ q) ∼(p ∨ q) means (∼p) ∧ (∼ q) ∼(p ⇒q) ⇔ ∼(∼p ∨ q) ⇔ p ∧ (∼ q ) ∼(p ⇔ q) ⇔ ∼[(p ⇒q) ∧ (q ⇒p)] ⇔ ∼(p ⇒q) ∨ ∼(q ⇒p) ⇔ [p ∧ (∼ q)] ∨ [q ∧ (∼ p)]

Nitin Oke Yaa Jatharpeth Road Akola 444005 Ph 0724 2420561

• RULES  ∧ is associative p ∧ ( q ∧ r) ⇔ (p ∧ q) ∧ r  ∨ is associative p ∨ ( q ∨ r) ⇔ (p ∨ q) ∨ r  ∨ is distributive over ∧ , Means r ∨ (p ∧ q) ⇔ (r ∨ p) ∧ (r ∨ q)  ∧ is distributive over ∨, Means r ∧ (p ∨ q) ⇔ (r ∧ p) ∨ (r ∧ q)

Nitin Oke Yaa Jatharpeth Road Akola 444005 Ph 0724 2420561

 A compound statement which is always true is called as Tautology  A compound statement which is always false is called as contradiction or fallacy  A compound statement which is neither contradiction nor tautology is called as contingency  A compound statement which is obtained by replacing ∨ by ∧ and ∧ by ∨ is called dual  Principle of duality means if a statement is tautology or fallacy then its dual also remains tautology or contradiction Nitin Oke Yaa Jatharpeth Road Akola 444005 Ph 0724 2420561

Switching circuits p

q

p ∧ q

p p ∨ q q

Nitin Oke Yaa Jatharpeth Road Akola 444005 Ph 0724 2420561