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Karnataka Class 11 Commerce Maths UNIT V MATHEMATICAL REASONING

Karnataka Class 11 Commerce Maths UNIT V MATHEMATICAL REASONING

Karnataka Class 11 Commerce Maths UNIT V MATHEMATICAL REASONING

Karnataka Class 11 Commerce Maths UNIT V MATHEMATICAL REASONING : If an object is either black or white, and if it is not black, then logic leads us to the conclusion that it must be white. Observe that logical reasoning from the given hypotheses can not reveal what “black” or “white” mean, or why an object can not be both. Infact, logic is the study of general patterns of reasoning, without reference to particular meaning or context.

Karnataka Class 11 Commerce Maths UNIT V MATHEMATICAL REASONING – syllabus

  • Definition of proposition and problems,
  • Logical connectives, compound proposition, problems,
  • Quantifiers, negation, consequences of implication-contrapositive and converse ,
  • problems , proving a statement by the method of contradiction by giving counter example.

Karnataka Class 11 Commerce Maths UNIT V MATHEMATICAL REASONING

A statement is a sentence which is either true or false, but not both simultaneously.

Note: No sentence can be called a statement if

(i) It is an exclamation

(ii) It is an order or request

(iii) It is a question

(iv) It involves variable time such as ‘today’, ‘tomorrow’, ‘yesterday’ etc.

(v) It involves variable places such as ‘here’, ‘there’, ‘everywhere’ etc.

(vi) It involves pronouns such as ‘she’, ‘he’, ‘they’ etc.

Example 1

(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“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.

Karnataka Class 11 Commerce Maths UNIT V MATHEMATICAL REASONING

Simple statements

A statement is called simple if it can not be broken down into two or more statements.

Example 2 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.

Compound statements

A compound statement is the one which is made up of two or more simple statements.

Example 3 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.

Karnataka Class 11 Commerce Maths UNIT V MATHEMATICAL REASONING

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.

Karnataka Class 11 Commerce Maths UNIT V MATHEMATICAL REASONING

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 4

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 5

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 6

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.

Karnataka Class 11 Commerce Maths UNIT V MATHEMATICAL REASONING

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 7

Form the disjunction of the following simple statements: p : The sun shines. q : It rains.

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 8

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.

Karnataka Class 11 Commerce Maths UNIT V MATHEMATICAL REASONING

Negation

An assertion that a statement fails or denial of a statement is called the negation of the statement. The negation of a statement is generally formed by introducing the word “not” at some proper place in the statement or by prefixing the statement with “It is not the case that” or It is false that”. The negation of a statement p in symbolic form is written as “~ p”.

Example 9

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 10

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.

Karnataka Class 11 Commerce Maths UNIT V MATHEMATICAL REASONING

Negation of compound statements

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 11

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).

Karnataka Class 11 Commerce Maths UNIT V MATHEMATICAL REASONING

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 12

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.

Karnataka Class 11 Commerce Maths UNIT V MATHEMATICAL REASONING

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

Karnataka Class 11 Commerce Maths UNIT V MATHEMATICAL REASONING

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 13

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 14

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”.

Karnataka Class 11 Commerce Maths UNIT V MATHEMATICAL REASONING

Contrapositive of a conditional statement

The statement “(~ q) → (~ p)” is called the contrapositive of the statement p → q

Example 15

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”.

Karnataka Class 11 Commerce Maths UNIT V MATHEMATICAL REASONING

Converse of a conditional statement

The conditional statement “q → p” is called the converse of the conditional statement “ p → q ”

Example 16

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.”

Karnataka Class 11 Commerce Maths UNIT V MATHEMATICAL REASONING

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 17

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 18

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.

Karnataka Class 11 Commerce Maths UNIT V MATHEMATICAL REASONING

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.

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