Question

# In the below options, the correct truth table of $p\wedge \left( \sim q \right)$ is:(a) pq$p\wedge \left( \sim q \right)$TTFTFFFTTFFF(b) pq$p\wedge \left( \sim q \right)$TTFTFTFTFFFF(c) pq$p\wedge \left( \sim q \right)$TTFTFFFTFFFT(d) pq$p\wedge \left( \sim q \right)$TTTTFFFTFFFF

Hint: In the above problem, we have to find the correct truth table of $p\wedge \left( \sim q \right)$ from the given options. Now, we are going to show the meaning of the symbols given in this expression $p\wedge \left( \sim q \right)$. $''\wedge ''$ means multiplication and $\sim q$ means opposite of the given value of q like if q is true then $\sim q$ is false. The concept that we are going to use in the multiplication table is that the result of the multiplication of a true statement with a true statement is true and the result of the multiplication of a false statement with a false statement is false. The result of the multiplication of a true statement with a false statement is a false statement.
We are asked to find the truth table for $p\wedge \left( \sim q \right)$.
The symbol $''\wedge ''$ means multiplication and $\sim q$ means opposite of the given value of q like if q is true then $\sim q$ is false.
The way of multiplication of p and q is done as follows:
The result of the multiplication of a true statement with a true statement is true.
The result of the multiplication of a false statement with a false statement is false.
The result of the multiplication of a true statement with a false statement is false.
Now, let us symbolize true as “T” and false as “F” and constructing the truth table of $p\wedge \left( \sim q \right)$ we get,
 p q $p\wedge \left( \sim q \right)$ T T F T F T F T F F F F

The explanation of the above table is that:
In the first row, the multiplication of the true statement of p with the false statement of $\sim q$ is false.
You might think that q is given as true then why we have multiplied false because we have to multiply $\sim q$ not q and $\sim q$ is the opposite of q. Like if q is true then $\sim q$ is false.
Similarly, you can explain the truth values of the remaining rows.
Hence, the correct option is (b).

Note: The place where the mistake could happen in this problem is that you forgot to consider $\sim$ in $\sim q$ and just use the truth value of q, not $\sim q$. For e.g., in the below, we have shown the correct truth table.

 p q $p\wedge \left( \sim q \right)$ T T F T F T F T F F F F

Now, take the second row of this table, if you forgot to consider $\sim q$ then the truth value of q is false, and multiplying false with the true statement of p we get a false statement and which is not the correct value so this is the place where the tendency of making mistakes is pretty high so make sure you won’t make this mistake.