Answer

Verified

449.4k+ views

Hint: Here, we will solve the given problem by considering each statements truth value and verify which compound statement is T i.e., Tautology.

Complete step-by-step answer:

i. $p \vee q$

Given,

The truth value of p is T and the truth value of q is F.

Now, we need to find the truth value of $p \vee q$ i.e.., “or” connectivity of statements of p and q

As, we know in the “or” connectivity if either of the statements p and q is $T$ then the compound statement $p \vee q$ will also be T

\[ p \vee q \\

T \vee F \\

T \\ \]

Hence, the truth value of $p \vee q$ is T i.e.., Tautology.

ii. $ \sim p \vee q$

Given,

The truth value of p is T and the truth value of q is F.

Therefore the truth value of ~p is F

Now, we need to find the truth value of $ \sim p \vee q$ i.e.., “or” connectivity of statements of negation (p) and q

\[ \Rightarrow \sim p \vee q \\

\Rightarrow F \vee F \\

\Rightarrow F \\ \]

Hence, the truth value of $ \sim p \vee q$ is F i.e.., Contradiction.

iii. $p \vee ( \sim q)$

Given,

The truth value of p is T and the truth value of q is F.

Therefore the truth value of ~q is T.

Now, we need to find the truth value of $p \vee ( \sim q)$ i.e.., “or” connectivity of statements of p and negation (q).

As, we know in the “or” connectivity if either of the statements is T then the compound statement will also be T

\[ \Rightarrow p \vee \sim q \\

\Rightarrow T \vee T \\

\Rightarrow T \\ \]

Hence, the truth value of $p \vee ( \sim q)$ is T i.e.., Tautology.

iv. $p \wedge ( \sim q)$

Given,

The truth value of p is T and the truth value of q is F.

Therefore the truth value of ~q is T.

Now, we need to find the truth value of $p \wedge ( \sim q)$ i.e.., “and” connectivity of statements of p and negation (q).

As we know in the “and” connectivity if the truth value of each statement is T then only the compound statement’s truth value will be T.

\[\Rightarrow p \wedge \sim q \\

\Rightarrow T \wedge T \\

\Rightarrow T \\ \]

Hence, the truth value of $p \wedge ( \sim q)$ is T i.e.., Tautology.

Therefore, (i), (iii), (iv) statements have the truth value T i.e.., Tautology.

Hence, the correct option for the given question is ‘B’.

Note: In solving the problems on truth values of statements if the connectivity is “or” then the truth value of the compound statement will be true if either of the two statements is true and if the connectivity is “and” then both of the statements should be true for the compound statement to be true.

Complete step-by-step answer:

i. $p \vee q$

Given,

The truth value of p is T and the truth value of q is F.

Now, we need to find the truth value of $p \vee q$ i.e.., “or” connectivity of statements of p and q

As, we know in the “or” connectivity if either of the statements p and q is $T$ then the compound statement $p \vee q$ will also be T

\[ p \vee q \\

T \vee F \\

T \\ \]

Hence, the truth value of $p \vee q$ is T i.e.., Tautology.

ii. $ \sim p \vee q$

Given,

The truth value of p is T and the truth value of q is F.

Therefore the truth value of ~p is F

Now, we need to find the truth value of $ \sim p \vee q$ i.e.., “or” connectivity of statements of negation (p) and q

\[ \Rightarrow \sim p \vee q \\

\Rightarrow F \vee F \\

\Rightarrow F \\ \]

Hence, the truth value of $ \sim p \vee q$ is F i.e.., Contradiction.

iii. $p \vee ( \sim q)$

Given,

The truth value of p is T and the truth value of q is F.

Therefore the truth value of ~q is T.

Now, we need to find the truth value of $p \vee ( \sim q)$ i.e.., “or” connectivity of statements of p and negation (q).

As, we know in the “or” connectivity if either of the statements is T then the compound statement will also be T

\[ \Rightarrow p \vee \sim q \\

\Rightarrow T \vee T \\

\Rightarrow T \\ \]

Hence, the truth value of $p \vee ( \sim q)$ is T i.e.., Tautology.

iv. $p \wedge ( \sim q)$

Given,

The truth value of p is T and the truth value of q is F.

Therefore the truth value of ~q is T.

Now, we need to find the truth value of $p \wedge ( \sim q)$ i.e.., “and” connectivity of statements of p and negation (q).

As we know in the “and” connectivity if the truth value of each statement is T then only the compound statement’s truth value will be T.

\[\Rightarrow p \wedge \sim q \\

\Rightarrow T \wedge T \\

\Rightarrow T \\ \]

Hence, the truth value of $p \wedge ( \sim q)$ is T i.e.., Tautology.

Therefore, (i), (iii), (iv) statements have the truth value T i.e.., Tautology.

Hence, the correct option for the given question is ‘B’.

Note: In solving the problems on truth values of statements if the connectivity is “or” then the truth value of the compound statement will be true if either of the two statements is true and if the connectivity is “and” then both of the statements should be true for the compound statement to be true.

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Why Are Noble Gases NonReactive class 11 chemistry CBSE

Let X and Y be the sets of all positive divisors of class 11 maths CBSE

Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE

Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE

Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Write a letter to the principal requesting him to grant class 10 english CBSE

Summary of the poem Where the Mind is Without Fear class 8 english CBSE

Difference Between Plant Cell and Animal Cell

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Change the following sentences into negative and interrogative class 10 english CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE