Tiêu đề XI - maths - chapter 14 - MATHEMATICAL REASONING (174-186).pdf ✅

174 NARAYANAGROUP MATHEMATICAL REASONING JEE-MAIN-SR-MATHS VOL-I MATHEMATICAL REASONING In a mathematical language, there are two kinds of reasoning inductive and deductive. We have already discussed the inductive reasoning in the context of mathematical induction. In this chapter, we shall discuss some fundamentals of deductive reasoning.  Sentence:We communicate our ideas or thoughts with the help of sentences in particular languages. Following types of sentences are normally used. (i) Assertive Sentence : A sentence that makes an assertion is called an ‘assertive sentence or a declarative sentence’. Eg: New Delhi is the capital of India (ii) Imperative Sentence: A sentence that expresses a request or a command is called an imperative sentence. Eg: please give me a glass of water (iii)Exclamatory Sentence: A sentence that expresses some strong feeling is called an exclamatory sentence. Eg:Oh God!what a beautiful scene (iv) Interrogative sentence : A sentence that asks some question is called an interrogative sentence. Eg: To which state do you belong? (v) Optative sentence: A sentence that expresses a wish is called an optative sentence. Eg:God bless you.  Statement (or) Proposition : A sentence is called a mathematically acceptable statement if it is either true(T) or false(F) but not both. Eg: Natural numbers are always positive Statements are usually dentoted by the letters p,q,r,......etc.  The truthness or falsity of a statement is called its truth value. Truthness of a statement is denoted by T. while its falsity is denoted by F.  True statements : Eg: (i) 2012 is a leap year, (ii)The sum of all interior angles of a triangle is 1800 .  False statements : Eg: (i) All prime numbers are odd integers. (ii) Two plus two is five.  Not a statement : Eg:(i) Mathematics is difficult. (ii) Tomorrow is Sunday.  Simple Statement : Any statement or proposition whose truth value does not explicitly depend on another statement is called a simple statement. Eg: Sun rises in the east. Its truth value is T  Compound statement: A statement which is made up of two or more simple statements using the connectives “and( )”, “or( )”, “implies()”, “if and only if( )” etc... is called a compound statement. In this case each statement is called a component statement. Eg: This book is for mathematics and its target is Jee-mains  Sub-Statement: The simple statements which form a compound statement are known as its sub- statements or components or constituents.  If p, q, r ...... are sub-statements of a compound statement S then the compound statement can be written as S p q r  , , .... .  Compound statement is that its truth value is completely determined by the truth values of the sub-statements together with the way in which they are connected to form the compound statement.  Open Statement:A sentence which contains one or more variables such that when certain values are given to the variable it becomes a statement, is called an open statement. Eg: “He is a great man” is an open sentence because in the sentence “He” can be replaced by any person.  Eg:Which of the following statement is/are open statement(s)? (1) Ram eats a mango. (2) Krishna goes to school (3) He lives in India (4) Anil and Anuj are good friends. Sol: In a given options, only option (3) is an open statement, because in this sentence ‘he’ can be specifed to any person.  Truth Table : A table that shows the relationship between the truth value of compound statement, S(p,q,r,...) and the truth values of its substatements p,q,r,...etc.., is called the truth tables of statement S. (i) For a single statement p, number of rows=21 =2 P T F SYNOPSIS
NARAYANAGROUP 175 JEE-MAIN-SR-MATHS VOL-I MATHEMATICAL REASONING (ii) For two statements p and q, number of rows=22 =4 p q T T T F F F T F (iii) For the three statements p,q,r, Number of Rows =23 =8 p q r T F F T F T T T F T T T F F F F F T F T F F T T  Note: If a compound statement has simply n substatements, then there are 2n rows representing logical possibilities. W.E-1: If there are 6 simple statements, then for making a table, find number of rows Sol: We know that, if compound statements has n substatements, then there are 2n rows in a table. Here, n=6  Total number of rows = 26 =64  Basic logical connectives or logical operators : Two or more statements are combined to form a compound statement by using symbols. These symbols are called logical connectives. Logical connectives are given below. Connective and or If ....... then Symbol    or  Nature of the compound statement formed by using the connective Conjunction Disjunction Implication or conditional If and only if (iff) not   or  Equivalence or Bi-conditional or Bi-implication Negation  Negation( ~) : The process of forming the contradictory of a given statement is called negation.  If p is a statement, then the negation of p is also a statement denoted by ~p.  Eg :- p: New Delhi is a city, then ~p: It is false that New Delhi is a city. (or) New Delhi is not a city  Negation Truth table : p T F F T ~p  Conjunction(  ): Any two simple statements can be connected by the word ‘and’ to form a compound statement called the conjunction of the original statements. Let p and q be two statements. The conjunction of p and q is denoted by p q  , read as p and q  Truth table for Conjunction : p T T F F T F T F T F F F q p q   Eg:- p: Two is an even number. q: Two is a prime number. p q  : Two is an even number and a prime number.  Disjunction ( Alternation )( ) : Any two statements can be connected by the word ‘or’ to form a compound statement called the ‘disjunction’ of the original statements. Let p and q be two statements the disjunction of p and q is denoted by p q  , read as p or q.  Truth table for disjunction : p T T F F T F T F T T T F q p q 
176 NARAYANAGROUP MATHEMATICAL REASONING JEE-MAIN-SR-MATHS VOL-I Eg:p: Two is an even number. q: Two is a prime number. p q  : Two is an even number or a prime number.  Conditional (or) Implication: Two statements connected by the connective phrase ‘if .... then’ give rise to a compound statement which is known as an implication or a conditional statement. If p and q are two statements forming the implication of ‘if p then q’ then the implication denoted by ' p q  or ' p q  . p is called the ‘antecedent’ and q is called ‘consequent’.  p q  read as p implies q, q if p, p is sufficient for q, q is necessary for p.  p q  is the statement that is false when p is true and q is false and true otherwise.  Truth table for Conditional : p T T F F T F T F T F T T q p q   Eg:- p: An integer is a multiple of 9. q: An integer is a multiple of 3. p q  :If an integer is a multiple of 9 then it is a multiple of 3.  Bi-implication   or : A statement is a biconditional statement if it is the conjunction of two conditional statements one converse to the other. If p, q are two statements, then the compound statement p q  and q p  is called a biconditional statement or an equivalence and is denoted by p q  ,  p q q p       .  p q  is the statement that is the true when p and q have the same truth value and otherwise false.  Truth table for Bi-implication : p T T F F T F T F T F F T q p q   Eg:- p: A number is divisible by 3. q: Sum of the digits of a number is divisible by 3. p q  :A number is divisible by 3 if and only if the sum of its digits is divisible by 3.  Converse: Let p,q be two statements. “If q then p” is called the converse of “ if p then q”. Thus the converse of p qis q p   .  Inverse : Let p,q be two statements. “ if ~p then ~q” is called the inverse of “ if p then q”. Thus the inverse of p q  is ~ p ~ q .  Contrapositive :The statement   q p  is called the contrapositive of p q  .  Eg:- p: x is an even integer. q: 2 x is divisible by 4. (i) p q  : If x is even integer then 2 x is divisible by 4. (ii) q p  : 2 x is divisible by 4 then x is even. (iii)   p q  : If x is not even integer then 2 x is not divisible by 4. (iv)   q p  : if 2 x is not divisible by 4 then x is not an even integer.  Converse, inverse and contra positive of a conditional : Suppose p, q are two statements such that p q  then i) Converse is q p  ii) Inverse is   p q  iii) Contra positive is   q p   Truth Table : p q Conditional p q  Converse q p  Inverse ~p ~q  Contra positive ~q ~p  T T F F T F T F T F T T T T F T T T F T T F T T  Tautology, Contradiction : (i) A compound statement that is always true is called a tautology. (ii) A compound statement that is always false is called a contradiction or fallassy. Eg:- Let p be a statement  Truth table : P ~P p ~P  p ~P  T F F T T T F F
NARAYANAGROUP 177 JEE-MAIN-SR-MATHS VOL-I MATHEMATICAL REASONING (i) p  ~ p is a tautology (ii) p  ~ p is a contradiction  Logical Equivalence: The statements p and q are called logically equivalent if they have the same entries in the last column of the truth tables.  Eg:- (i) ~  p q   and (~ p)  (~ q) are logically equivalent. (ii) p q  and ~ p q  are logically equivalent. (iii) p q q p      (iv)    p q p q     p q p q  ~ (p q)  ~ p ~ q ~ p ~q  T T F F T F T F T T T F F F F T F F T T F T F T F F F T p q ~ p (~ p) q  p q  T T F F T F T F F F T T T F T T T F T T  The phrases, ‘for all’, ‘for some’, ‘for no’, ‘for every’ and ‘there exists atleast one’ convey the idea of quantity and refer to some specific collection of numbers or objects. these phrases quantify the variable in open sentences. they are called quantifiers.  The quantifier ‘for all’ or ‘for every’ is called the universal quantifier and is denoted by  . The quantifier ‘some’ or ‘there exists atleast one’ is called existantial quantifier and is denoted by the symbol .  Algebra of statements : 1. Idempotent laws : For any statement p, (i) p p p   (ii) p p p   2. Commutative laws: For any statements p and q (i) p q q p    (ii) p q q p    3. Associative laws: For any three statements p,q,r (i)  p q r p q r         (ii)  p q r p q r         4. Distributive Laws: For any three statements p,q,r, (i) p q r p q p r             (ii) p q r p q p r             5. DeMorgan’s laws: If p and q are two statements, then (i) ~ p q   (~p) (~q) (ii) ~ p q   (~p) (~q) 6. Identity laws: If t and c denote a tautology and a contradiction respectively, then for any statement p, (i) p t p   (ii) p c p   (iii) p t t   (iv) p c c   7. Complement laws: For any statement p, (i) p  ( p) t (ii) p  ( p)c 8. Law of contrapositive : For any two statements p and q, (i) p q   q p 9. Involution Laws: For any statement p, we have ( p)  p . 1. If a sentence can be judged to be true or false, but not both then it is called 1) an open sentence 2) a statement 3) a tautology 4) a contradiction 2. The truthfulness or falsity of a statement is called its 1) negation 2) converse 3) inverse 4) truth value 3. Which of the following is a proposition 1) Logic is an interesting subject 2) He is very talented 3) I am a lion 4) A triangle is a circle and 10 is a prime number 4. Which of the following is not a proposition 1) 3 is a prime 2) Mathematics is interesting 3) 5 is an even integer 4) 2 is irrational 5. Denial of a statement is called is 1) negation 2) converse 3) inverse 4) truth value 6. The symbolic form of “p or q” is 1) p q  2) p q  3) p q  4) p q  7. The symbolic form of “ p and q” is 1) p q  2) p q  3) p q  4) p q  8. The symbolic form of “(either p) or (not p )” is 1) p p ~  2) p p ~  3) p p  ~  4) p p  ~  C.U.Q