Mathematics For 12th Class students

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 ex

Hello friends, Welcome to my blog mathstips4u. In my last post we have seen converse, Inverse and contrapositive of an implication and its examples. Some of the examples were left as exercise for you. That will be covered in this post. If you not still read that post, please read that post before watching this video. In this post we are going to learn what is meant by tautology, contradiction and contingency and their examples. Tautology (t): A statement pattern which is true for all the combinations of the truth values of its component statements, is called a tautology. e.g. p v ~p p ~p p v ~p T F T F T T   Contradiction(c): A statement pattern which is false for all the combinations of the truth values of its component statements, is called a contradiction. e.g. p Ʌ ~p p ~p p Ʌ ~p T F F F T F It is obvious that the negation of a tautology is a contra

Hello friends, Welcome to my blog mathstips4u. In my last post we have seen logical equivalence and some of their examples. In this post we are going to learn converse , inverse and contrapositive of an implication and some of their examples. If p → q is an implication, then there arises following three implications 1.      q → p is called converse of p → q 2.        ~ p →    ~q is called inverse of p → q 3.         ~ q →    ~ p is called contrapositive of p → q We shall see the truth table of converse , inverse and contrapositive of an implication. p q ~ p ~ q p → q conditional q → p converse ~ p →~q inverse ~ q → ~ p contrapositive T T F F T T T T T F F T F T T F F T T F T F F T F F T T T T T T 1 2 3 4 5