Question

How many reflexive relations there in a set of n + 1 elements?

• A. $$n(n+1)$$
• B. $$n^2$$
• C. $$2^n$$
• D. $$2^{n(n+1)}$$

Answer 1

Answer:

$$2^{n(n+1)}$$

• Let the given set, say set X, have N = n + 1 elements.

• The total number of possible ordered pairs in the Cartesian product X × X is N × N = (n + 1) × (n + 1) = (n + 1)².

• For a relation to satisfy the reflexive property, all the N 'diagonal' ordered pairs (e.g., (x₁,x₁), (x₂,x₂), ..., (x_N,x_N)) must be included in the relation. There is only one choice for each of these N pairs (they are mandatory members of the relation).

• The number of remaining ordered pairs in X × X (which are not diagonal elements) is calculated by subtracting the number of diagonal elements from the total number of pairs: (n + 1)² - (n + 1).

• Factoring out (n + 1) from this expression, we get (n + 1) * ((n + 1) - 1) = (n + 1) * n = n(n + 1).

• For each of these n(n + 1) non-diagonal pairs, there are exactly two independent choices: either the pair is included in the relation, or it is not.

• Since each choice is independent, the total number of ways to form reflexive relations is 2 multiplied by itself n(n + 1) times, leading to the formula 2^{n(n + 1)}.