Direct method of Validation

Q.

Let p be the statement “x is an irrational number”, q be the statement “y is a transcendental number”, and r be the statement “x is a rational number if and only if y is a transcendental number”.
Statement-1 : $risequivalenttoeitherqorp$
Statememt-2 : $risequivalentto(p\leftrightarrow \sim q)$

1. Statement – 1 is false, Statement – 2 is true
2. Statement – 1 is true, Statement – 2 is true; Statement -2 is not a correct explanation for Statement -1

3. Statement – 1 is true, Statement – 2 is true; Statement – 2 is a correct explanation for Statement -1

4. Statement – 1 is true. Statement – 2 is false

Q.

Are the following pairs of statements negations of each other:

(i) The number x is not a rational number. The number x is not an irrational number.

(ii) The number x is a rational number. The number x is an irrational number.

Q.

Check the validity of the statements given below by the method given against it

(i) p : The sum of an irrational number and a rational number is irrational (by contradiction method)

(ii) q : If n is a real number with n > 3, then $n^2>9$ (by contradiction method)

Q.

Let p be the statement “x is an irrational number”, q be the statement “y is a transcendental number”, and r be the statement “x is a rational number if and only if y is a transcendental number”.
Statement-1 : $risequivalenttoeitherqorp$
Statememt-2 : $risequivalentto(p\leftrightarrow \sim q)$

1. Statement – 1 is false, Statement – 2 is true

2. Statement – 1 is true, Statement – 2 is true; Statement – 2 is a correct explanation for Statement -1

3. Statement – 1 is true, Statement – 2 is true; Statement -2 is not a correct explanation for Statement -1

4. Statement – 1 is true. Statement – 2 is false

Q.

Find the truth value of the following statement : "If x, y $\in Z$ are such that x and y are odd, then xy is odd."

1. True

2. False