Write the component statements of the following compound statements an

1.	Write the component statements of the following compound statements andcheck whether the compound statement is true or false.(i) 57 is divisible by 2 or 3.(ii) 24 is a multiple of 4 and 6.(iii) All living things have two eyes and two legs.(iv) 2 is an even number and a prime number.
Answer» (i) Here component statements are: p: 57 is divisible by 2. [false] q: 57 is divisible by 3. [true] Given compound statement is of the form ‘pvq’. Since, the statement ‘pvq’ has the truth value T whenever either p or q or both have the truth value T. So, it is true statement as 57 is divisible by 3. (ii) Here component statements are: p: 24 is multiple of 4. q: 24 is multiple of 6. Given compound statement is of the form ‘p ^ q’ Since, the statement ‘p A q’ has the truth value T whenever bothp and q have the truth value T. So, it is a true statement as 24 is divisible by 4 and 6. (iii) Here component statements are: p: All living things have two eyes. [false] q: All living things have two legs. [false] Given compound statement is of the form ‘p ^q’ It is a false statement. Since ‘p ^ q’ has truth value F whenever either p or q or both have the truth value F (iv) Here component statements are: p: 2 is an even number. [true] q: 2 is a prime number. [true] Given compound statement is of the form ‘p ^ q’. It is a true statement. Since ‘p ^ q’ has truth value T whenever both p and q or both have the truth value T.

Write the component statements of the following compound statements andcheck whether the compound statement is true or false.(i) 57 is divisible by 2 or 3.(ii) 24 is a multiple of 4 and 6.(iii) All living things have two eyes and two legs.(iv) 2 is an even number and a prime number.

Answer»

(i) Here component statements are:

p: 57 is divisible by 2. [false]

q: 57 is divisible by 3. [true]

Given compound statement is of the form ‘pvq’.

Since, the statement ‘pvq’ has the truth value T whenever either p or q or both have the truth value T.

So, it is true statement as 57 is divisible by 3.

(ii) Here component statements are: p: 24 is multiple of 4. q: 24 is multiple of 6.

Given compound statement is of the form ‘p ^ q’

Since, the statement ‘p A q’ has the truth value T whenever bothp and q have the truth value T.

So, it is a true statement as 24 is divisible by 4 and 6.

(iii) Here component statements are:

p: All living things have two eyes. [false]

q: All living things have two legs. [false]

Given compound statement is of the form ‘p ^q’

It is a false statement. Since ‘p ^ q’ has truth value F whenever either p or q or both have the truth value F

(iv) Here component statements are:

p: 2 is an even number. [true]

q: 2 is a prime number. [true]

Given compound statement is of the form ‘p ^ q’.

It is a true statement. Since ‘p ^ q’ has truth value T whenever both p and q or both have the truth value T.