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Question

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p: I am topper.

q: I worked hard.

$

{\text{A}}{\text{. p}} \leftrightarrow {\text{q}} \\

{\text{B}}{\text{. p}} \vee {\text{q}} \\

{\text{C}}{\text{. p}} \wedge {\text{q}} \\

{\text{D}}{\text{. p}} \to {\text{q}} \\

$

Answer

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Hint: Here, we will be proceeding by simply using all the four logic symbols mentioned in the options with the two statement variables p and q given in the problem to see which one of these symbolic forms gives the same statement as given in the problem.

Complete step-by-step answer:

The statement is “I am topper and I worked hard”. This statement needs to be represented in the symbolic form using two different statement variables p and q with the help of some logic symbol.

Given, p: I am topper and q: I worked hard.

If we see the options, four different logic symbols are used. Let us observe these one by one.

First logic symbol between p and q is $ \leftrightarrow $ which stands for equivalence. As we know that if there are two statement variables A and B then ${\text{A}} \leftrightarrow {\text{B}}$ means “A if and only if B” . This is the symbolic form.

So, symbolic form \[{\text{p}} \leftrightarrow {\text{q}}\] means “I am topper if and only if I worked hard” which is not the same as the given statement.

Second logic symbol between p and q is $ \vee $ which stands for disjunction. As we know that if there are two statement variables A and B then ${\text{A}} \vee {\text{B}}$ means “A or B” . This is the symbolic form.

So, symbolic form \[{\text{p}} \vee {\text{q}}\] means “I am topper or I worked hard” which is not the same as the given statement.

Third logic symbol between p and q is $ \wedge $ which stands for conjunction. As we know that if there are two statement variables A and B then ${\text{A}} \wedge {\text{B}}$ means “A and B” . This is the symbolic form.

So, the symbolic form \[{\text{p}} \wedge {\text{q}}\] means “I am topper and I worked hard” which is the same as the given statement.

Fourth logic symbol between p and q is $ \to $ which stands for implication. As we know that if there are two statement variables A and B then ${\text{A}} \to {\text{B}}$ means “If A then B” . This is the symbolic form.

So, the symbolic form \[{\text{p}} \to {\text{q}}\] means “If I am topper then I worked hard” which is not the same as the given statement.

Clearly, the symbolic form of the statement: “I am topper and I worked hard” is \[{\text{p}} \wedge {\text{q}}\] where two statement variables are p as “I am topper” and q as “I worked hard”.

Hence, option C is correct.

Note: In these types of problems, we represent all the given symbolic forms in the options into statements which will be formed with the help of statement variables p and q given in the problem. Here, there is no need for truth tables because we have to just understand the symbolic form of various logic symbols.

Complete step-by-step answer:

The statement is “I am topper and I worked hard”. This statement needs to be represented in the symbolic form using two different statement variables p and q with the help of some logic symbol.

Given, p: I am topper and q: I worked hard.

If we see the options, four different logic symbols are used. Let us observe these one by one.

First logic symbol between p and q is $ \leftrightarrow $ which stands for equivalence. As we know that if there are two statement variables A and B then ${\text{A}} \leftrightarrow {\text{B}}$ means “A if and only if B” . This is the symbolic form.

So, symbolic form \[{\text{p}} \leftrightarrow {\text{q}}\] means “I am topper if and only if I worked hard” which is not the same as the given statement.

Second logic symbol between p and q is $ \vee $ which stands for disjunction. As we know that if there are two statement variables A and B then ${\text{A}} \vee {\text{B}}$ means “A or B” . This is the symbolic form.

So, symbolic form \[{\text{p}} \vee {\text{q}}\] means “I am topper or I worked hard” which is not the same as the given statement.

Third logic symbol between p and q is $ \wedge $ which stands for conjunction. As we know that if there are two statement variables A and B then ${\text{A}} \wedge {\text{B}}$ means “A and B” . This is the symbolic form.

So, the symbolic form \[{\text{p}} \wedge {\text{q}}\] means “I am topper and I worked hard” which is the same as the given statement.

Fourth logic symbol between p and q is $ \to $ which stands for implication. As we know that if there are two statement variables A and B then ${\text{A}} \to {\text{B}}$ means “If A then B” . This is the symbolic form.

So, the symbolic form \[{\text{p}} \to {\text{q}}\] means “If I am topper then I worked hard” which is not the same as the given statement.

Clearly, the symbolic form of the statement: “I am topper and I worked hard” is \[{\text{p}} \wedge {\text{q}}\] where two statement variables are p as “I am topper” and q as “I worked hard”.

Hence, option C is correct.

Note: In these types of problems, we represent all the given symbolic forms in the options into statements which will be formed with the help of statement variables p and q given in the problem. Here, there is no need for truth tables because we have to just understand the symbolic form of various logic symbols.

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