Quantifiers and Quantified statements

In my last post we have seen Tautology, contradiction and contingency with some examples.

In this post we are going to learn quantifiers and quantified statements and some of their examples with solution.

There will be a questions in HSC board exam. For 1 or 2 marks.

In mathematics we come across the statements such as

1) “for all”,  x Є R, x^2＞ or = 0  and   2)  “there exist “, x Є N such that x + 5 = 9.

In these statement the phrases “for all” and “there exist “are called quantifiers and these above statements are called quantified statements.

i.e. An open sentence with a quantifier becomes a statement and is called a quantified statement.

In mathematical logic there are two quantifiers

1) Universal Quantifiers (ꓯ):

“for all” x or “for every” x is called universal quantifier and we use the symbol ‘ꓯ’ to denote this.

The statement 1) in above  is written like ꓯ x Є R, x^2＞ or = 0.  

2) Existential quantifiers(ⱻ):

The phrase “there exist “is called existential quantifier which indicates the at least one element exists that satisfies a certain condition and the

symbol used is ‘ⱻ’.

The second statement is written symbolic form as

ⱻ x Є N, ⱻ such that x + 5 = 9.

Now we shall see the examples

Ex. 1. Use quantifiers to convert each of the following open sentences defined on N, into a true statement.

i) x + 4 = 5,   ii) x^2＞ 0,  iii) x + 3 ＜ 6

Solution:   i) ⱻ x Є N, such that x + 4 = 5.

 It is a true statement, since x = 1ЄN, satisfies x + 4 = 5.

ii)x^2 ＞0, ꓯ x Є N. It is a true statement, since the square of every natural number is positive.

iii) ⱻ x Є N, such that x + 3 ＜ 6. It is a true statement, for x = 1 or 2 Є N, satisfy x + 3 ＜ 6.

Ex.2) If A = {3,4, 6, 8} determined the truth of each of the following.

i) ⱻ x Є A, such that x + 4 = 7.

Clearly x = 3 Є A, satisfies x + 4 = 7.  It is true statement. T

ii) ꓯ x Є A, x + 4 ＜10

Since x = 6 and 8 Є A, do not satisfy x + 4 ＜10, the given statement is false. F

This is all about Quantifiers and Quantified statements.

In my next post we are going to learn what is meant by duality in logic.