Understanding Quantifiers and Quantified Statements

Quantifiers and Quantified statements.<br />In my last video we have seen Tautology, contradiction and contingency with some examples.<br />In this video we are going to learn quantifiers and quantified statements and some of their examples with solution.<br />There will be a questions in HSC board exam. For 1 or 2 marks.<br />In mathematics we come across the statements such as <br />1) “for all”, x Є R, x^2 ＞ or = 0 and 2) “there exist “, x Є N such that x + 5 = 9.<br />In these statement the phrases “for all” and “there exist “are called quantifiers and these above statements are called quantified statements.<br />i.e. An open sentence with a quantifier becomes a statement and is called a quantified statement.<br />In mathematical logic there are two quantifiers <br />1) Universal Quantifiers (ꓯ): <br />“for all” x or “for every” x is called universal quantifier and we use the symbol ‘ꓯ’ to denote this.<br />The statement 1) in above is written like ꓯ x Є R, x^2 ＞ or = 0. <br />2) Existential quantifiers(ⱻ):<br />The phrase “there exist “is called existential quantifier which indicates the at least one element exists that satisfies a certain condition and the <br />symbol used is ‘ⱻ’.<br />The second statement is written symbolic form as <br />ⱻ x Є N, ⱻ such that x + 5 = 9.<br />Now we shall see the examples<br />Ex. 1. Use quantifiers to convert each of the following open sentences defined on N, into a true statement.<br />i) x + 4 = 5, ii) x^2 ＞ 0, iii) x + 3 ＜ 6<br />Solution: i) ⱻ x Є N, such that x + 4 = 5.<br /> It is a true statement, since x = 1ЄN, satisfies x + 4 = 5.<br />ii) x^2 ＞0, ꓯ x Є N. It is a true statement, since the square of every natural number is positive.<br />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.<br />Ex.2) If A = {3,4, 6, 8} determined the truth of each of the following.<br />i) ⱻ x Є A, such that x + 4 = 7.<br />Clearly x = 3 Є A, satisfies x + 4 = 7. It is true statement. T<br />ii) ꓯ x Є A, x + 4 ＜10<br />Since x = 6 and 8 Є A, do not satisfy x + 4 ＜10, the given statement is false. F<br />This is all about Quantifiers and Quantified statements.<br />In my next video we are going to learn what is meant by duality in logic.<br />Visit my website:<br />https://mathstips4u.blogspot.com/2019/07/quantifiers-and-quantified-statements.html<br />