** ****Q.1:**

**(i). Every real number is not complex number and every rational number is a real number.**

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

**(iii). The sand easily heats up due to the sun but does not cool down easily at night**

**(iv). The roots for the equation x + 10 = 3x ^{2} are x = 3 and x = 2**

** **

**Sol:**

**(i)** Here ‘and’ is a connecting word. Here component statements will be:

**a:** **Every real number is not complex.**

**b:** **Every rational number are real.**

**(ii) ** Here, ‘or’ is a connecting word. Here component statements will be:

**a: Square of any integer is negative.**

**b: Square of any integer is positive.**

**(iii)** Here ‘but’ is connecting word. Here component statements will be:

**a: The sand heats up easily due to sun.**

**b: The sand dose not cool down easily at night.**

**(iv)** Here ‘and’ is the connecting word. Here component statements will be:

**a: The roots for the equation x + 10 = 3x ^{2} are x = 3**

**b: The roots for the equation x + 10 = 3x ^{2} are x = 2**

**Q.2: Write negation for the statements after identifying the quantifier for the statements**

**(i). There exits one number that is equal to the square of the number**

**(ii). For every number that is real ‘x’, x < x + 1 **

**(iv). There exist one capital for each state of India.**

**Sol:**

**(i)** The quantifier will be “There exist” and negation for the statement is:

**There doesn’t exist any number that is equal to the square of the number.**

**(ii)** The quantifier will be “ For every” and negation for the statement is:

**There exist a number x that is not less than x + 1**

**(iii)** The quantifier will be “There exist” and negation for the statement is:

**There exist one state that has no capital.**

**Q.3: Check if the following statements are negation for each other. Justify your answer**

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

**(ii). There exist real numbers x, y such that y + x = x + y**

** **

**Sol:**

The negation for **statement** **(i)** will be:

There exist real numbers x, y such that y + x ≠ x + y, which is not **statement (ii)**.

**So the statements are not negation for each other.**

**Q.4: State if the “Or” in the statements is inclusive or exclusive. Justify the answer**

**(i) Moon sets or sun rises**

**(ii) You must have ration card or passport for applying a driving license.**

**(iii) Integers are negative or positive**

** **

**Sol:**

**(i) “or” is exclusive as it is impossible for moon to set and sun to rise together.**

**(ii) “or” is inclusive as one can have both passport and ration card for applying a driving license.**

**(iii) “or” is exclusive as integers can’t be negative and positive.**