example-1Multiple choice
- (i)The sentence ‘New Delhi is in India; is true. So it is a statement.
- (ii)The sentence “Every rectangle is a square” is false. So it is a statement.
- (iii)The sentence “Close the door” can not be assigned true or false (Infact, it is a command). So it can not be called a statement.
- (iv)The sentence MATHEMATICAL REASONING 247 “How old are you?” can not be assigned true or false (In fact, it is a question). So it is not a statement. (v) The truth or falsity of the sentence “x is a natural number” depends on the value of x. So it is not considered as a statement. However, in some books it is called an open statement. Note: Truth and falisity of a statement is called its truth value. 14.1.2 Simple statements A statement is called simple if it can not be broken down into two or more statements.
example-2Short answer
The statements “2 is an even number”, “A square has all its sides equal” and “ Chandigarh is the capital of Haryana” are all simple statements. 14.1.3 Compound statements A compound statement is the one which is made up of two or more simple statements.
example-3Long answer
The statement “11 is both an odd and prime number” can be broken into two statements “11 is an odd number” and “11 is a prime number” so it is a compound statement. Note: The simple statements which constitutes a compound statement are called component statements. 14.1.4 Basic logical connectives There are many ways of combining simple statements to form new statements. The words which combine or change simple statements to form new statements or compound statements are called Connectives. The basic connectives (logical) conjunction corresponds to the English word ‘and’; disjunction corresponds to the word ‘or’; and negation corresponds to the word ‘not’. Throughout we use the symbol ‘∧’ to denote conjunction; ‘∨’ to denote disjunction and the symbol ‘~’ to denote negation. Note: Negation is called a connective although it does not combine two or more statements. In fact, it only modifies a statement. 14.1.5 Conjunction If two simple statements p and q are connected by the word ‘and’, then the resulting compound statement “p and q” is called a conjunction of p and q and is written in symbolic form as “p ∧ q”.
example-4Short answer
Form the conjunction of the following simple statements: p: Dinesh is a boy. q: Nagma is a girl. Solution The conjunction of the statement p and q is given by p∧q: Dinesh is a boy and Nagma is a girl.
example-5True / False
Translate the following statement into symbolic form “Jack and Jill went up the hill.” Solution The given statement can be rewritten as “Jack went up the hill and Jill went up the hill” Let p : Jack went up the hill and q : Jill went up the hill. Then the given statement in symbolic form is p ∧ q. Regarding the truth value of the conjunction p ∧ q of two simple statements p and q, we have (D1) : The statement p ∧ q has the truth value T (true) whenever both p and q have the truth value T. (D2) : The statement p ∧ q has the truth value F (false) whenever either p or q or both have the truth value F.
example-6Multiple choice
Write the truth value of each of the following four statements:
- (i)Delhi is in India and 2 + 3 = 6.
- (ii)Delhi is in India and 2 + 3 = 5.
- (iii)Delhi is in Nepal and 2 + 3 = 5.
- (iv)Delhi is in Nepal and 2 + 3 = 6. Solution In view of (D1) and (D2) above, we observe that statement (i) has the truth value F as the truth value of the statement “2 + 3 = 6” is F. Also, statement (ii) has the truth value T as both the statement “Delhi is in India” and “2 + 3 = 5” has the truth value T. Similarly, the truth value of both the statements (iii) and (iv) is F. 14.1.6 Disjunction If two simple statements p and q are connected by the word ‘or’, then the resulting compound statement “p or q” is called disjunction of p and q and is written in symbolic form as “p ∨ q”.
example-7Long answer
Form the disjunction of the following simple statements: p: The sun shines. q: It rains. MATHEMATICAL REASONING 249 Solution The disjunction of the statements p and q is given by p ∨ q : The sun shines or it rains. Regarding the truth value of the disjunction p ∨ q of two simple statements p and q, we have (D3) : The statement p ∨ q has the truth value F whenever both p and q have the truth value F. (D4) : The statement p ∨ q has the truth value T whenever either p or q or both have the truth value T.
example-8Multiple choice
Write the truth value of each of the following statements:
- (i)India is in Asia or 2 + 2 = 4.
- (ii)India is in Asia or 2 + 2 = 5.
- (iii)India is in Europe or 2 + 2 = 4.
- (iv)India is in Europe or 2 + 2 = 5. Solution In view of (D3) and (D4) above, we observe that only the last statement has the truth value F as both the sub-statements “India is in Europe” and “2 + 2 = 5” have the truth value F. The remaining statements (i) to (iii) have the truth value T as at least one of the sub-statements of these statements has the truth value T.
example-9Long answer
Write the negation of the statement p : New Delhi is a city. Solution The negation of p is given by ~ p : New Delhi is not a city or ~ p : It is not the case that New Delhi is a city. or ~ p : It is false that New Delhi is a city. Regarding the truth value of the negation ~ p of a statement p, we have (D5) : ~ p has truth value T whenever p has truth value F. (D6) : ~ p has truth value F whenever p has truth value T.
example-10Multiple choice
Write the truth value of the negation of each of the following statements:
- (i)p : Every square is a rectangle.
- (ii)q : The earth is a star.
- (iii)r : 2 + 3 < 4 Solution In view of (D5) and (D6), we observe that the truth value of ~p is F as the truth value of p is T. Similarly, the truth value of both ~q and ~r is T as the truth value of both statements q and r is F. 14.1.8 Negation of compound statements 14.1.9 Negation of conjunction Recall that a conjunction p ∧ q consists of two component statements p and q both of which exist simultaneously. Therefore, the negation of the conjunction would mean the negation of at least one of the two component statements. Thus, we have (D7) : The negation of a conjunction p ∧ q is the disjunction of the negation of p and the negation of q. Equivalently, we write ~ (p ∧ q) = ~ p ∨ ~ q
example-11Multiple choice
Write the negation of each of the following conjunctions:
- (a)Paris is in France and London is in England.
- (b)2 + 3 = 5 and 8 < 10. Solution (a) Write p : Paris is in France and q : London is in England. Then, the conjunction in (a) is given by p ∧ q. Now ~ p : Paris is not in France, and ~ q : London is not in England. Therefore, using (D7), negation of p ∧ q is given by ~ ( p ∧ q) = Paris is not in France or London is not in England. (b) Write p : 2 + 3 = 5 and q : 8 < 10. Then the conjunction in (b) is given by p ∧ q. Now ~ p : 2 + 3 ≠ 5 and ~ q : 8 </ 10. Then, using (D7), negation of p ∧ q is given by – ( p ∧ q) = (2 + 3 ≠ 5 ) or (8 </ 10) MATHEMATICAL REASONING 251 14.1.10 Negation of disjunction Recall that a disjunction p ∨ q is consisting of two component statements p and q which are such that either p or q or both exist. Therefore, the negation of the disjunction would mean the negation of both p and q simultaneously. Thus, in symbolic form, we have (D8) : The negation of a disjunction p ∨ q is the conjunction of the negation of p and the negation of q. Equivalently, we write ~ (p ∨ q) = ~ p ∧ ∼ q
example-12Multiple choice
Write the negation of each of the following disjunction :
- (a)Ram is in Class X or Rahim is in Class XII.
- (b)7 is greater than 4 or 6 is less than 7. Solution (a) Let p : Ram is in Class X and q : Rahim is in Class XII. Then the disjunction in (a) is given by p ∨ q. Now ~p: Ram is not in Class X. ~q : Rahim is not in Class XII. Then, using (D8), negation of p ∨ q is given by ~ (p ∨ q) : Ram is not in Class X and Rahim is not in Class XII. (b) Write p : 7 is greater than 4, and q : 6 is less than 7. Then, using (D8), negation of p ∨ q is given by ~ (p ∨ q) : 7 is not greater than 4 and 6 is not less than 7. 14.1.11 Negation of a negation As already remarked the negation is not a connective but a modifier. It only modifies a given statement and applies only to a single simple statement. Therefore, in view of (D5) and (D6), for a statement p, we have (D9) : Negation of negation of a statement is the statement itself. Equivalently, we write ~ ( ~ p) = p 14.1.12 The conditional statement Recall that if p and q are any two statements, then the compound statement “if p then q” formed by joining p and q by a connective ‘if then’ is called a conditional statement or an implication and is written in symbolic form as p → q or p ⇒ q. Here, p is called hypothesis (or antecedent) and q is called conclusion (or consequent) of the conditional statement (p ⇒ q): Remark The conditional statement p ⇒ q can be expressed in several different ways. Some of the common expressions are : (a) if p, then q (b) q if p
- (c)p only if q
- (d)p is sufficient for q (e) q is necessary for p. Observe that the conditional statement p → q reflects the idea that whenever it is known that p is true, it will have to follow that q is also true.
example-13Multiple choice
Each of the following statements is also a conditional statement.
- (i)If 2 + 2 = 5, then Rekha will get an ice-cream.
- (ii)If you eat your dinner, then you will get dessert.
- (iii)If John works hard, then it will rain today.
- (iv)If ABC is a triangle, then ∠ A + ∠ B + ∠ C = 180°.
example-14Short answer
Express in English, the statement p → q, where p : it is raining today q:2+3>4 Solution The required conditional statement is “If it is raining today, then 2 + 3 > 4” 14.1.13 Contrapositive of a conditional statement The statement “(~ q) → (~ p)” is called the contrapositive of the statement p → q
example-15Multiple choice
Write each of the following statements in its equivalent contrapositive form:
- (i)If my car is in the repair shop, then I cannot go to the market.
- (ii)If Karim cannot swim to the fort, then he cannot swim across the river. Solution (i) Let “p : my car is in the repair shop” and “q : I can not go to the market”. Then, the given statement in symbolic form is p → q. Therefore, its contrapositive is given by ~ q → ~ p. Now ~ p : My car is not in the repair shop. and ~ q : I can go to the market Therefore, the contrapositive of the given statement is “If I can go to the market, then my car is not in the repair shop”. (ii) Proceeding on the lines of the solution of (i), the contrapositive of the statement in (ii) is “If Karim can swim across the river, then he can swim to the fort”. MATHEMATICAL REASONING 253 14.1.14 Converse of a conditional statement The conditional statement “q → p” is called the converse of the conditional statement “ p → q ”
example-16Multiple choice
Write the converse of the following statements
- (i)If x < y, then x + 5 < y + 5
- (ii)If ABC is an equilateral triangle, then ABC is an isosceles triangle Solution (i) Let p:x<y q:x+5<y+5 Therefore, the converse of the statement p → q is given by “If x + 5 < y + 5, then x < y (ii) Converse of the given statement is “If ABC is an isosceles triangle, then ABC is an equilateral triangle.” 14.1.15 The biconditional statement If two statements p and q are connected by the connective ‘if and only if’ then the resulting compound statement “p if and only if q” is called a biconditional of p and q and is written in symbolic form as p ↔ q.
example-17Short answer
Form the biconditional of the following statements: p : One is less than seven q : Two is less than eight Solution The biconditional of p and q is given by “One is less than seven, if and only if two is less than eight”.
example-18Multiple choice
Translate the following biconditional into symbolic form: “ABC is an equilateral triangle if and only if it is equiangular”. Solution Let p : ABC is an equilateral triangle and q : ABC is an equiangular triangle. Then, the given statement in symbolic form is given by p ↔ q. 14.1.16 Quantifiers Quantifieres are the phrases like ‘These exist’ and “for every”. We come across many mathematical statement containing these phrases. For example – Consider the following statements p : For every prime number x, x is an irrational number. q : There exists a triangle whose all sides are equal. 14.1.17 Validity of statements Validity of a statement means checking when the statement is true and when it is not true. This depends upon which of the connectives, quantifiers and implication is being used in the statement. (i) Validity of statement with ‘AND’ To show statement r : p ∧ q is true, show statement ‘p’ is true and statement ‘q’ is true. (ii) Validity of statement with ‘OR’ To show statement r : p ∨ q is true, show either statement ‘p’ is true or statement ‘q’ is true. (iii) Validity of statement with “If-then” To show statement r : “If p then q is true”, we can adopt the following methods:
- (a)Direct method : Assume p is true and show q is true, i.e., p ⇒ q.
- (b)Contrapositive method : Assume ~ q is true and show ~ p is true, i.e., ~ q ⇒ ~ p.
- (c)Contradiction method : Assume that p is true and q is false and obtain a contradiction from assumption.
- (d)By giving a counter example : To prove the given statement r is false we give a counter example. Consider the follwoing statement. “r : All prime numbers are odd”. Now the statement ‘r’ is false as 2 is a prime number and it is an even number. 14.1.18 Validity of the statement with “If and only If” To show the statement r : p if and only if q is true, we proceed as follows: Step 1 Show if p is true then q is true. Step 2 Show if q is true then p is true.
example-1Multiple choice
Which of the following statements are compound statements
- (i)“2 is both an even number and a prime number”
- (ii)“9 is neither an even number nor a prime number”
- (iii)“Ram and Rahim are friends” MATHEMATICAL REASONING 255 Solution (i) The given statement can be broken into two simple statements “2 is an even number” and “2 is a prime number” and connected by the connective ‘and’ (ii) The given statement can be broken into two simple statements “9 is not an even number” and “9 is not a prime number” and connected by the connective ‘and’ (iii) The given statement can not be broken into two simple statements and hence it is not a compound statement.
example-2Multiple choice
Identify the component statements and the connective in the following compound statements.
- (a)It is raining or the sun is shining.
- (b)2 is a positive number or a negative number. Solution (a) The component statements are given by p : It is raining q : The sun is shining The connective is “or” (b) The component statements are given by p : 2 is a positive number q : 2 is a negative number The connective is ‘or’
example-3Multiple choice
Translate the following statements in symbolic form
- (i)2 and 3 are prime numbers
- (ii)Tigers are found in Gir forest or Rajaji national park. Solution (i) The given statement can be rewritten as “2 is a prime number and 3 is a prime number”. Let p : 2 is a prime number q : 3 is a prime number Then the given statement in symbolic form is p ∧ q. (ii) The given statement can be rewritten as “Tigers are found in Gir forest or Tigers are found in Rajaji national park” Let p : Tigers are found in Gir forest q : Tigers are found in Rajaji national park. Then the given statement in symbolic form is p ∨ q.
example-4Multiple choice
Write the truth value of each of the following statements.
- (i)9 is an even integer or 9 + 1 is even.
- (ii)2 + 4 = 6 or 2 + 4 = 7
- (iii)Delhi is the capital of India and Islamabad is the capital of Pakistan.
- (iv)Every rectangle is a square and every square is a rectangle. (v) The sun is a star or sun is a planet. Solution In view of (D1), (D2), (D3) and (D4), we observe that only statement (iv) has truth value F as the first component statement namely “every rectangle is a square” is false. Further, in statements (i), (ii) and (v) atleast one component statement is true. Therefore, these statements have truth value T. Also, truth value of statement (iii) is T as both the component statements are true.
example-5Short answer
Write negation of the statement “Everyone who lives in India is an Indian” Solution Let p : Everyone who lives in India is an Indian. The negation of this statement is given by ~ p : It is false that everyone who lives in India is an Indian. ~ p : Everyone who lives in India is not an Indian.
example-6Multiple choice
Write the negation of the following statements :
- (a)p : All triangles are equilateral triangles.
- (b)q : 9 is a multiple of 4.
- (c)r : A triangle has four sides. Solution (a) We have It is false that all triangles are equilateral triangles MATHEMATICAL REASONING 257 ~ p : Threre exists a triangle which is not an equilateral triangles. ~ p : Not all triangles are equilateral triangles (b) ~ q : 9 is not a multiple of 4. (c) ~ r : It is false that the triangle has four sides. ~ r : A triangle has not four sides.
example-7Multiple choice
Write the negation of the following statements :
- (i)Suresh lives in Bhopal or he lives in Mumbai.
- (ii)x + y = y + x and 29 is a prime number. Solution (i) Let p : Suresh lives in Bhopal and q : Suresh lives in Mumbai Then the disjunction in (i) is given by p ∨ q. Now ~ p : Suresh does not live in Bhopal. ~ q : Suresh does not live in Mumbai. Therefore, using (D8), negation of p ∨ q is given by ~ ( p ∨ q ) : Suresh does not live in Bhopal and he does not live in Mumbai. (ii) Let p : x + y = y + x and q : 29 is a prime number. Then the conjunction in (ii) is given by p ∧ q. Now ~ p : x + y ≠ y + x and ~ q : 29 is not a prime number. Therefore, using (D7), negation of p ∧ q is given by, ~ ( p ∧ q) : x + y ≠ y + x or 29 is not a prime number.
example-8Multiple choice
Rewrite each of the following statements in the form of conditional statements :
- (i)Mohan will be a good student if he studies hard.
- (ii)Ramesh will get dessert only if he eats his dinner.
- (iii)When you sing, my ears hurt.
- (iv)A necessary condition for Indian team to win a cricket match is that the selection committee selects an all-rounder. (v) A sufficient condition for Tara to visit New Delhi is that she goes to the Rashtrapati Bhawan. Solution (i) The given statement is of the form “q if p”, where p : Mohan studies hard. q : He will be a good student. It is an equivalent form (Remark (b) 14.1.12) of the statement “if p then q”. So the equivalent formulation of the given statement is “If Mohan studies hard, then he will be a good student”. (Here, note that in p he is replaced by Mohan and in q Mohan is replaced by he) (ii) The given statement is of the form “p only if q” which is an equivalent form (Remark (c) 14.1.12) of the statement “if p then q”. So, the equivalent formulation of the given statement is: “If Ramesh eats his dinner, then he will get dessert” (iii) Here ‘when’ means the same as ‘if’ and so the equivalent formulation of the given statements is: “If you sing, then my ears hurt” (iv) The given statement is of the form “q is necessary for p” where p : Indian team wins a cricket match q : The selection committee selects an all-rounder which is an equivalent form (Remark (e) 14.1.12) of “if p then q”. So the equivalent formulation of the given statement is “If the teams wins a cricket match then selection committee selects an all rounder. (v) The given statement is of the form “p is sufficient for q” where p : Tara goes to Rashtrapati Bhawan q : She visits New Delhi which is an equivalent form (Remark (d) 14.1.12) of “if p, then q”, so the equivalent formulation of the given statement is “If Tara goes to Rashtrapati Bhawan, then she visits New Delhi”. MATHEMATICAL REASONING 259
example-9Short answer
Express in English, the statement p → q, where p : It is raining today q:2+3>4 Solution The conditional statement is “If it is raining today, then 2 + 3 > 4”.
example-10Short answer
Translate the following statements in symbolic form: If x = 7 and y = 4” then x + y = 11. Solution Let p : x = 7 and y = 4 and q : x + y = 11 Then the given statement is symbolic form is p → q
example-11Short answer
Form the biconditional of the following statements : p : Today is 14th of August q : Tomorrow is Independence day Solution The biconditional p ↔ q is given by “Today is 14th of August if and only if tomorrow is Independence Day”.
example-12Short answer
Translate the following biconditional into symbolic form: “ABC is an equilateral triangle if and only if its each interior angle is 60°” Solution Let p : ABC is an equilateral triangle and q : Each interior angle of triangle ABC is 60° Then the given statement in symbolic form is p ↔ q.
example-13Multiple choice
Identify the quantifiers and write the negation of the following statements
- (i)There exists a number which is equal to its square.
- (ii)For all even integers x, x2 is also even.
- (iii)There exists a number which is a multiple of 6 and 9. Solution (i) The quantifier is “there exists” and the negation is “There does not exist a number which is equal to its square” (ii) The quantifier is “for all” and the negation is “There exists an even integer x such that x2 is not even” (iii) The quantifier is “there exists” and the negation is “There does not exist a number which is a multiple of both 6 and 9”.
example-14Long answer
Show that the following statement is true. p : For any real numbers x, y if x = y, then 2x + a = 2y + a when a ∈ Z. Solution We prove the statement ‘p’ is true by contrapositive method and by Direct Method. Direct Method for any real number x, y given x=y ⇒ 2x = 2y ⇒ 2x + a = 2y + a for some a ∈ Z. Contrapositve Method The contrapositive statement of ‘p’ is “For any real numbers x, y if 2x + a ≠ 2y + a, where a ∈ Z, then x ≠ y. Given 2x + a ≠ 2y + a ⇒ 2x ≠ 2y ⇒ x≠y
example-15Multiple choice
Check the validity of the statements
- (i)r : 100 is a multiple of 4 and 5.
- (ii)s : 60 is a multiple of 3 or 5. Solution (i) Let p : r ∧ s where r : “100 is a multiple of 4” is true s : “100 is a multiple of 5” is true Hence p is true. (ii) Let q : r V s, where r : “60 is a multiple of 3”, is true. s : “60 is a multiple of 5”, is true. Hence q is true.
example-16Multiple choice
Which of the following is a statement?
- (A)Roses are black.
- (B)Mind your own business.
- (C)Be punctual.
- (D)Do not tell lies. MATHEMATICAL REASONING 261 Solution (A) is the correct answer as the sentences in (B), (C) and (D) are neither true nor false. Infact all these sentences are advices.
example-17Multiple choice
The negation of the statement “It is raining and weather is cold.” is
- (A)It is not raining and weather is cold.
- (B)It is raining or weather is not cold.
- (C)It is not raining or weather is not cold.
- (D)It is not raining and weather is not cold. Solution (C) is the correct answer as it satisfies (D7). The options (A), (B) and (D) do not satisfy (D7).
example-18Multiple choice
Which of the following is the converse of the statement ? “If Billu secure good marks, then he will get a bicycle.”
- (A)If Billu will not get bicycle, then he will not secure good marks.
- (B)If Billu will get a bicycle, then he will secure good marks.
- (C)If Billu will get a bicycle, then he will not secure good marks.
- (D)If Billu will not get a bicycle, then he will secure good marks. Solution (B) is the correct answer since the statement q → p is the converse of the statement p → q.