Question 1: Find the component statements of the following compound statements:

(i) The sky is blue and the grass is green.

The component statements of the given compound statement are:

1) The sky is blue.

2) The grass is green.

(ii) The earth is round or the sun is cold.

The component statements of the given compound statement are:

1) The earth is round.

2) The sun is cold.

(ii) All rational numbers are real and all real numbers are complex.

The component statements of the given compound statement are:

1) All rational numbers are real.

2) All real numbers are complex.

(iv) 25 is a multiple of 5 and 8.

The component statements of the given compound statement are:

1) 25 is a multiple of 5.

2) 25 is a multiple of 8. $\\$

Question 2: For each of the following statements, determine whether an inclusive “OR” or exclusive “OR” is used. Give reasons for your answer.

(i) Students can take Hindi or Sanskrit as their third language.

Exclusive OR is used because students can opt for either Hindi or Sanskrit as third language.

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

Inclusive OR is used because a person can have both passport as well as voter registration card.

(ii) A lady gives birth to a baby boy or a baby girl.

Exclusive OR is used because a lady can give birth to a baby who is either a boy or a girl.

(iv) To apply for a driving license, you should have a ration card or a passport.

Inclusive OR is used because a person could have both a ration card and a passport. $\\$

Question 3: Write the component statements of the following compound statements and check whether
the compound statement is true or false:

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

The component statements of the given compound statement are:

1) To enter into a public library children, need an identity card from the school.

2) To enter into a public library children, need a letter from the school authorities.

The compound statement is true because both component statements are true.

(ii) All rational numbers are real and all real numbers are not complex.

The component statements of the given compound statement are:

1) All rational numbers are real.

2) All real numbers are not complex.

The compound statement is false because all real numbers are complex. The connective used is AND. So, even if one component statement is false, the compound statement is false.

(iii) Square of an integer is positive or negative.

The component statements of the given compound statement are:

1) Square of an integer is positive.

2) Square of an integer is negative.

The compound statement is true because the first statement is true. Since the connective used is OR and one of the component statements is true, the compound statement is true.

(iv) $x = 2$ and $x = 3$ are the roots of the equation $3x^2 -x-10 = 0.$

The component statements of the given compound statement are:

1) $x = 2$ is the root of the equation $3x^2 -x-10 = 0.$

2) $x = 3$ is the root of the equation $3x^2 -x-10 = 0.$

The connective used is AND. So, both component statements must be true for the compound statement to be true. The statement $x = 3$ is the root of the equation $3x^2 -x-10 = 0$ is false. Therefore, the compound statement is false.

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

The component statements of the given compound statement are:

1) The sand heats up quickly in the sun

2) The sand does not cool down fast at night.

The connective used is AND. For the compound statement to be true, both component statements must be true. The second component statement ” The sand does not cool down fast at night.” is false. Sand cools down fast at night. Therefore, the compound statement is false. $\\$

Question 4: Determine whether the following compound statements are true or false:

(i) Delhi is in India and $2 + 2 = 4$

The component statements of the given compound statement are:

1) Delhi is in India.

2) $2 + 2 = 4$

True. Both component statements are true and the connective is AND.

(ii) Delhi is in England and $2 + 2 =4$

The component statements of the given compound statement are:

1) Delhi is in England.

2) $2 + 2 = 4$

False. The first compound statement “Delhi is in England” is false. Since the connective is AND and one component statement is false, the compound statement is false.

(iii) Delhi is in India and $2 + 2 = 5$

The component statements of the given compound statement are:

1) Delhi is in India.

2) $2 + 2 = 5$

False. The second compound statement $2 + 2 = 5$ is false. Since the connective is AND and one component statement is false, the compound statement is false.

(iv) Delhi is in England and $2 + 2 =5$

The component statements of the given compound statement are:

1) Delhi is in England.

2) $2 + 2 = 5$

False. Both component statements are false. Therefore, the compound statement is false.