Question

The symbolic form of the statement: “I am topper and I worked hard”, if
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 1

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.