Chapter 31: Mathematical Reasoning  – Exercise 31.3

Mathematical Reasoning – Exercise – 31.3 – Q.1

(i)

The components of the compound statement:

The sky is blue and the grass is green.

are :

p : The sky is blue

q : The grass is green

(ii)

The components of the compound statement:

The earth is round or the sun is cold.

are:

p : The earth is round

q : The sun is cold.

(iii)

The components of the compound statement:

All rational numbers are real and all real numbers are complex.

are:

p : All rational numbers are real

q: All real numbers are complex

(iv)

The components of the compound statement:

25 is a multiple of and 8.

are:

p :25 is multiple of 5

q: 25 is multiple of 8.

Mathematical Reasoning – Exercise – 31.3 – Q.2

(i)

In the statement:

Students can take Hindi or Sanskrit as their third language.

an exclusive "OR" is used because

A student cannot take both Hindi and Sanskrit as the third language.

(ii)

In the statement:

To entry a country, you need a passport or a voter registration card.

an inclusive "OR' is used because

Since a person can have both a passport and a voter registration card to enter a country.

(iii)

In the statement:

A lady gives birth to a baby boy or a baby girl.

an exclusive "OR" is used because

A lady cannot give birth to a baby who is both a boy and a girl.

(iv)

 In the statement:

To apply for a driving licence, you should have a ration card or a passport.

an inclusive OR is used because

A person can have both a ration card and passport to apply for a driving licence.

Mathematical Reasoning – Exercise – 31.3 – Q.3

(i)

The component statements of the compound statement:

To enter into a public library children need an identity card from the school or a letter from the school authorities.

are

p : To get into a public library children need an identity card.

q : To get into a public library children need a letter from the school authorities.

We know that if p and q are true then p or q must also be true. Hence, the compound statement is true.

(ii)

The component statements of the compound statement:

All rational numbers are real and all real numbers are not complex.

are

p: All rational numbers are real.

q : All real numbers are not complex.

we know that p is true and q is false.

∴ The compound statement “p and q” is false.

(iii)

The component statements of the compound statement:

Square of an integer is positive or negative.

are

p : Square of an integer is positive.

q: Square of an integer is negative.

We know that if p and q are true then p or q must also be true.

Hence, the compound statement is true.

(iv)

The component statements of the compound statement:

x = 2 and x = 3 are the roots of the equation 3x² - x -10 = O.

are

p: x - 2 is a root of the equation 3x² – x - 10 = 0.

q: x = 3 is a root of the equation 3x² - x - 10 = O.

Here, p is true but q is false.

∴ The compound statement "p and q° is false.

(v)

The component statements of the compound statement:

The sand heats up quickly in the sun and does not cool down fast at night.

are

p: The sand heats up quickly in the sun.

q: The sand does not cool down fast at night.

The compound statement p and q is false, because p is true and q is false.

Mathematical Reasoning – Exercise – 31.3 – Q.4

The component statements of the compound statement:

Delhi is in India and 2 + 2 = 4.

are

p : Delhi is in India.

q : 2 + 2 = 4.

We know that if p and q are true then p or q must also be true.

Hence, the compound statement is true.

(ii)

The component statements of the compound statement: 

Delhi is in England and 2 + 2 = 4.

are

p : Delhi is in England

q : 2 + 2 = 4.

Here p is false and q is true. So, p and q must be false. Hence, the compound statement is false.

(iii)

The component statements of the compound statement:

Delhi is in India and 2 + 2 = 5.

are

p: Delhi is in India

q : 2 + 2 = 5.

Here, p is true and q is false. So, p and q must be false. Hence, the compound statement is false.

(iv)

The component statements of the compound statement:

Delhi is in England and 2 + 2 = 5.

are

p: Delhi is in England.

q : 2 + 2 = 5.

We know that if p and q are false then p and q must also be false. Hence, the compound statement is false.