NCERT Class 11-Math's: Chapter –14 Mathematical Reasoning Part 2

Glide to success with Doorsteptutor material for NSO : fully solved questions with step-by-step explanation- practice your way to success.

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 7: Form the disjunction of the following simple statements:

: The sun shines.

: It rains.

Solution The disjunction of the statements p and q is given by

: The sun shines or it rains.

Regarding the truth value of the disjunction pq of two simple statements p and q, we have

The statement has the truth value F whenever both p and q have the truth value F.

The statement 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

(ii) India is in Asia or

(iii) India is in Europe or

(iv) India is in Europe or

Solution: In view of and above, we observe that only the last statement has the truth value F as both the sub-statements “India is in Europe” and “” 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.

14.1.7 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 “”.

Example 9: Write the negation of the statement

New Delhi is a city.

Solution: The negation of p is given by

New Delhi is not a city

Or It is not the case that New Delhi is a city.

Or : It is false that New Delhi is a city.

Regarding the truth value of the negation of a statement p, we have

: p has truth value T whenever p has truth value F.

: 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) Every square is a rectangle.

(ii) The earth is a star.

(iii)

Solution: In view of and , we observe that the truth value of is F as the truth value of is T. Similarly, the truth value of both and is T as the truth value of both statements and r is F.

14.1.8 Negation of compound statements:

14.1.9 Negation of conjunction: Recall that a conjunction consists of two component statements and 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

The negation of a conjunction is the disjunction of the negation of and the negation of . Equivalently, we write

Example 11: Write the negation of each of the following conjunctions:

(a) Paris is in France and London is in England.

(b) and .

Solution:

(a) Write Paris is in France and q : London is in England.

Then, the conjunction in (a) is given by .

Now : Paris is not in France, and

: London is not in England.

Therefore, using , negation of is given by

Paris is not in France or London is not in England.

(b) Write and .

Then the conjunction in (b) is given by .

Now and .

Then, using , negation of is given by

14.1.10 Negation of disjunction: Recall that a disjunction pq 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

: The negation of a disjunction is the conjunction of the negation of p and the negation of . Equivalently, we write