Explain the types of quantifier.

Universal Quantifier:

Universal Quantifier is used for Universal Quantification and is denoted by ∀(for all). The Universal Quantification of P(x) is denoted by ∀xP(x) and ∀xP(x) is a proposition “p(x) is true for all the values of x in the universal set”. If x1,x2, x3,…, xn are the values in the universal set then the Universal Quantification ∀xP(x) is same as the conjunction P(x1) ˄ P(x2) ˄ P(x3) ˄ … ˄ P(xn).

Existential Quantifier:

Existential Quantifier is used for Existential Quantification and is denoted by ∃(there exist). The Existential Quantification of P(x) is denoted by ∃xP(x) and ∃xP(x) is a proposition “p(x) is true for at least one values of x in the universal set”. If x1,x2, x3,…, xn are the values in the universal set then the Existential Quantification ∃xP(x) is same as the disjunction P(x1) ∨ P(x2) ∨ P(x3) ∨ … ∨ P(xn).

What do you understand by Quantifier and Quantification?

Quantifier is the tool to make the propositional function a proposition.

Construction of the proposition from the predicates using quantifier is called quantification.

What do you mean by Duality principle?

Any logical expression that are deducible from the postulates of logical expressions remains valid if operators AND (˄) and OR (∨) are interchanged and the truth value T and F are interchanged, the new expression so formed is called Dual of the original expression.  If p is a proposition then its Dual is denoted by p*.

What is known as logical equivalence?

The compound proposition that has the same truth values in all possible cases is called logical equivalence. OR The proposition p and q is logical equivalent if p↔q is a tautology.

Eg: ~(p  ˄ q) and ~p ∨ ~q are logically equivalent.

p	q	¬p	¬q	p ˄ q	¬ (p ˄ q)	¬p ∨ ¬q	¬ (p ˄ q) ↔ ¬p ∨ ¬q
T	T	F	F	T	F	F	T
T	F	F	T	F	T	T	T
F	T	T	F	F	T	T	T
F	F	T	T	F	T	T	T

Since, ¬ (p ˄ q) ↔ ¬p ∨ ¬q is a tautology we can say that ¬ (p ˄ q) and ¬p ∨ ¬q are logically equivalent.