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Problem Summary

We are given:

1. A set $X = \{a, b, c, d, e, f\}$, which has 6 elements.
2. A symmetric relation $R$ on $X$, meaning that if $(x, y) \in R$, then $(y, x) \in R$ as well.
3. We need to find:
   • The number of symmetric relations on $X$.
   • The number of those symmetric relations that are also reflexive.

Solution Steps

1. Total Number of Pairs: Since $X$ has 6 elements, there are $6 \times 6 = 36$ possible ordered pairs $(x, y)$ for $x, y \in X$.

2. Symmetric Condition: To create a symmetric relation, if $(x, y) \in R$, then $(y, x)$ must also be in $R$. Therefore:

   • Diagonal Pairs: There are 6 diagonal pairs $(a, a), (b, b), (c, c), (d, d), (e, e), (f, f)$, which are naturally symmetric (since they are of the form $(x, x)$). We can choose whether or not to include each of these pairs independently.
   • Off-Diagonal Pairs: For each pair $(x, y)$ where $x \neq y$, we can either include both $(x, y)$ and $(y, x)$ in the relation or exclude both. There are $\binom{6}{2} = 15$ such pairs (since there are 15 unique pairs of distinct elements in a set of 6).

3. Counting Symmetric Relations:

   • Each of the 6 diagonal pairs can be independently included or excluded, giving $2^6 = 64$ choices.
   • Each of the 15 off-diagonal pairs also has 2 choices: include both $(x, y)$ and $(y, x)$ or exclude both. This gives $2^{15}$ choices for the off-diagonal pairs.
   • Thus, the total number of symmetric relations on $X$ is $2^6 \times 2^{15} = 2^{21} = 2,097,152$.

4. Counting Reflexive and Symmetric Relations:

   • For a relation to be reflexive, all diagonal pairs $(a, a), (b, b), (c, c), (d, d), (e, e), (f, f)$ must be included.
   • This means we have no choice for the 6 diagonal pairs—they must all be included.
   • We still have $2^{15}$ choices for the off-diagonal pairs (as we did for symmetric relations).
   • Therefore, the number of symmetric and reflexive relations is $2^{15} = 32,768$.

Final Answer

1. Total symmetric relations: $2^{21} = 2,097,152$.
2. Symmetric and reflexive relations: $2^{15} = 32,768$.

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Related Questions

1. What is the difference between reflexive, symmetric, and transitive properties in relations?
2. How many symmetric relations are there on a set with $n$ elements?
3. What changes if we also require the relation to be transitive?
4. How would the answer change if the set $X$ had 5 elements instead of 6?
5. Can a relation be symmetric but not reflexive? Provide examples.

Tip:

For any set with $n$ elements, a symmetric relation can be efficiently counted by separately considering diagonal and off-diagonal elements.