**Problem 1: For each of the following compound statements first identify the connecting words and then break it into component statements. **

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

**Solution:**

Here, the connecting word is “and”.

Now, the component statements of this compound statement are –

(a) All rational numbers are real.

(b) All real numbers are not complex.

**(ii) Square of an integer is positive or negative.**

**Solution: **

Here, the connecting word is “or”.

Now, the component statements of this compound statement are –

(a) Square of an integer is positive.

(b) Square of an integer is negative.

**(iii) The sand heats up quickly in the Sun and does not cool down fast at night.**

**Solution:**

Here, the connecting word is “and”.

Now, the component statements of this compound statement are –

(a) The sand heats up quickly in the Sun.

(b) The sand does not cool down fast at night.

**(iv)** **x = 2 and x = 3 are the roots of the equation 3x**^{2} – x – 10 = 0.

^{2}– x – 10 = 0.

**Solution:**

Here, the connecting word is “and”.

Now, the component statements of this compound statement are –

(a) x = 2 is the root of the equation 3x

^{2}– x – 10 = 0.(b) x = 3 is the root of the equation 3x

^{2}– x – 10 = 0.

**Problem 2: Identify the quantifier in the following statements and write the negation of the statements.**

**(i) There exists a number which is equal to its square.**

**Solution:**

Here, the quantifier is “There exists”.

Now, the negation of this statement will be: There does not exist a number which is equal to its square.

**(ii) For every real number x, x is less than x + 1. **

**Solution:**

Here, the quantifier is “For every”.

Now, the negation of this statement will be: For every real number x, x is not less than x + 1.

**(iii) There exists a capital for every state in India.**

**Solution:**

Here, the quantifier is “There exists”.

Now, the negation of this statement will be: There exists a state in India which does not have a capital.

**Problem 3: Check whether the following pair of statements are **the **negation of each other. Give reasons for your answer. **

**(a) x + y = y + x is true for every real numbers x and y.**

**(b) There exists real numbers x and y for which x + y = y + x.**

**Solution:**

Answer:The given pair of statements are not negation of each other.

Reason:To check if the given pair of statements are negation of each other, First of all we will find the negation of statement (a),

So, The negation of statement (a) : There exists real number x and y for which x + y ≠ y + x.

Since, The negation of statement (a) is not same as statement (b).

Thus, The given pair of statements are not negation of each other.

**Problem 4: State whether the “Or” used in the following statements is “exclusive “or” inclusive. Give reasons for your answer.**

**(i) Sun rises or Moon sets.**

**Solution:**

Answer:The “Or” used in this statement is exclusive.

Reason:We know that, It is not possible that Sun will rise and Moon will set together.

Since, It is not possible for both events to occur simultaneously.

Thus, The “Or” used in this statement is exclusive.

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

**Solution:**

Answer:The “Or” used in this statement is inclusive.

Reason:We know that, A person could have both a ration card and a passport to apply for a driving license.

Since, It is possible for both events to occur simultaneously.

Thus, The “Or” used in this statement is inclusive.

**(iii) All integers are positive or negative. **

**Solution:**

Answer:The “Or” used in this statement is exclusive.

Reason:We know that, A integer can not be both positive and negative.

Since, It is not possible for both events to occur simultaneously.

Thus, The “Or” used in this statement is exclusive.

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