Question

How many relations are there on a set with \(3\) elements?

• A. \(8\)
• B. \(128\)
• C. \(512\)
• D. \(256\)

Hint:

For a set with \(n\) elements: \[ \text{Number of relations} = 2^{n^2}. \] This is one of the most frequently asked formulas in discrete mathematics.

Answer 1

The Correct Option is C

Concept: A relation on a set \(A\) is any subset of \[ A\times A. \] If a set contains \(n\) elements, then \[ |A\times A|=n^2. \] Since every subset of \(A\times A\) is a relation, the total number of relations is \[ 2^{n^2}. \]

Step 1: Find the size of the Cartesian product.
The set contains \[ n=3 \] elements. Hence, \[ |A\times A|=3^2=9. \]

Step 2: Find the number of subsets.
Each relation is a subset of \(A\times A\). A set with \(9\) elements has \[ 2^9 \] subsets.

Step 3: Calculate the value.
\[ 2^9=512. \] Therefore the number of relations is \[ \boxed{512}. \] Hence option (C) is correct.