Compound Statement

Q. The statement $\sim (p\leftrightarrow \sim q)$ is

1. a tautology
2. a fallacy
3. equivalent to $p\leftrightarrow q$
4. equivalent to $\sim p\leftrightarrow q$

Q. If p : It rains today, q : I go to school, r : I shall meet any friends and s : I shall go for a movie, then which of the following is the proposition :
If it does not rain or if I do not go to school, then I shall meet my friend and go for a movie

1. $\sim (p\wedge q)\Rightarrow (r\wedge s)$
2. $\sim (p\wedge \sim q)\Rightarrow (r\wedge s)$
3. $\sim (p\wedge q)\Rightarrow (r\vee s)$
4. None of these

Q.

A statement which is made up of two or more statements is called ……

1. Compound statement

2. Subjective statement

3. Negative of statement

4. Ambiguous statement

Q.

The compound statement with ‘And’ is true if all its component statements are ……….

1. Different

2. False

3. Same

4. True

Q. A statement is a combination of two or more simple statements.

1. compound
2. elementary
3. fundamental

Q. Given below are two statements p : 25 is a multiple of 5. q: 25 is a multiple of 8. Write the compound statements connecting these two statements with “And” and “Or”. In both cases check the validity of the compound statement.

Q. 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. (ii) Square of an integer is positive or negative. (iii) The sand heats up quickly in the Sun and does not cool down fast at night. (iv) x = 2 and x = 3 are the roots of the equation 3 x 2 – x – 10 = 0.

Q.

Write the component statements of the compound statement and check whether the compound statement is true or false.

'A line is straight and extends indefinitely in both directions.'

Q.

The logically equivalent proposition of $p\iff q$ is

1. $(p\wedge q)\vee (p\wedge q)$
2. $(p\wedge q)\vee (q\Rightarrow p)$
3. $(p\Rightarrow q)\vee (q\Rightarrow p)$
4. $(p\wedge q)\Rightarrow (q\vee p)$

Q. Find the component statements of the following compound statements and check whether they are true or false. (i) Number 3 is prime or it is odd. (ii) All integers are positive or negative. (iii) 100 is divisible by 3, 11 and 5.

Q. Which of the following are examples of the null set?
(i) Set of odd natural numbers divisible by $$2$$
(ii) Set of even prime numbers
(iii) $$\left \{ x : x \text { is a natural numbers, $$x < 5$$ and $$x > 7}\right \}$$
(iv) $$\left \{ y : y \text { is a point common to any two parallel lines}\right \}$$

Q.

Write the component statement of the following compound statement and check whether the compound statement is true or false. All living things have two hands and two eyes."

Q. 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
(ii) Square of an integer is positive or negative
(iii) The sand heats up quickly in the Sun and does not cool down fast at night
(iv) $x=2$ and $x=3$ are the roots of the equation $3x^2-x-10=0$

Q. Find the component statements of the following compound statements:
(i) The sky is blue and the grass is green.
(ii) The earth is round or the sun is cold.
(iii) All rational numbers are real and all real numbers are complex.
(iv) 25 is a multiple of 5 and 8.

Q.

For each of the following compound statement 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.

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

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

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

Q. The Boolean expression $(p\wedge \sim q)\vee q\vee (\sim p\wedge q)$ is equivalent to

1. $p\vee \sim q$
2. $\sim p\wedge q$
3. $p\wedge q$
4. $p\vee q$

Q. P: Arjun is the fastest. Q: Azad is the captain. denotes the statement "Arjun is the fastest OR Azad is not the captain"

1. $P\wedge \sim Q$
2. $P\wedge Q$
3. $P\vee Q$
4. $P\vee \sim Q$

Q. Which of the following statements is not a tautology?

1. $p\wedge q\Rightarrow (\sim p)\vee q$
2. $(p\wedge q)\Rightarrow p$
3. $p\Rightarrow (p\vee q)$
4. $(p\vee q)\Rightarrow (p\vee (\sim q\left)\right)$

Q. The equivalent statement of $(p\leftrightarrow q)$ is

1. $(p\wedge q)\vee (p\vee q)$
2. $(\sim p\vee q)\wedge (q\vee \sim p)$
3. $(p\to q)\vee (q\to p)$
4. $(\sim p\wedge \sim q)\vee (p\vee q)$

Q. Determine whether the following compound statements are true or false:
(i) Delhi is in India and 2 + 2 = 4.
(ii) Delhi is in England and 2 + 2 = 4.
(iii) Delhi is in India and 2 + 2 = 5.
(iv) Delhi is in England and 2 + 2 =5.