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 â€śTâ€ť and false as â€śFâ€ť 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 wonâ€™t make this mistake.