ISRO CS 2023 Q.47: solution and answer

ISRO 2023 · CSQ.47 · 1 mark · MCQ

A truth table will need how many rows if there are n variables.

See the question in the paper

Answer and solution

ANSWER

A truth table on $n$ variables needs $2^{n}$ rows, option (c).

Each of the $n$ propositional variables can independently take one of two truth values, true or false.
A truth table lists every distinct assignment of values to the variables, one row per assignment.
By the multiplication rule, the number of assignments is $2 \times 2 \times \dots \times 2$ ( $n$ factors) $= 2^{n}$ .
Check: $n = 1$ gives 2 rows, $n = 2$ gives 4 rows, $n = 3$ gives 8 rows. The formula $2 n$ would give 2, 4, 6, so it fails at $n = 3$ . The formula $n$ gives 1, 2, 3, so it fails already at $n = 1$ .
So option (c), $2^{n}$ , is correct. Option (d) cannot be right because a correct option is listed.

KEY IDEA

Every variable doubles the number of possible assignments, so $n$ variables give $2^{n}$ rows.

WATCH OUT FOR

Writing $2 n$ or $n$ by extrapolating from the one-variable case, where 2 rows happen to equal $2 n$ . Test $n = 3$ : there are 8 rows, not 6 or 3.