Let `A={a,b,c}`. How may relation can be defined in the set ? How may of these are reflexive ?

Answer

To solve the problem, we need to determine the number of relations that can be defined on the set \( A = \{a, b, c\} \) and how many of these relations are reflexive.

### Step 1: Determine the number of relations on set \( A \)

1. **Identify the elements of the set**: The set \( A \) has 3 elements: \( a, b, c \).
2. **Find the Cartesian product \( A \times A \)**: The Cartesian product \( A \times A \) consists of all ordered pairs formed by taking one element from \( A \) and pairing it with another element from \( A \). Thus, we have:
\[ A \times A = \{(a, a), (a, b), (a, c), (b, a), (b, b), (b, c), (c, a), (c, b), (c, c)\} \]
This gives us a total of \( 3 \times 3 = 9 \) ordered pairs.
3. **Calculate the number of relations**: A relation on set \( A \) is any subset of \( A \times A \). The number of subsets of a set with \( n \) elements is given by \( 2^n \). Therefore, the number of relations is:
\[ 2^{|A \times A|} = 2^9 = 512 \]

### Step 2: Determine the number of reflexive relations

1. **Understand reflexive relations**: A relation is reflexive if every element is related to itself. For our set \( A \), this means that the pairs \( (a, a), (b, b), (c, c) \) must be included in any reflexive relation.
2. **Identify mandatory pairs**: The mandatory pairs for reflexivity are \( (a, a), (b, b), (c, c) \). This accounts for 3 pairs.
3. **Count remaining pairs**: The remaining pairs that can either be included or excluded from the relation are:
\[ (a, b), (a, c), (b, a), (b, c), (c, a), (c, b) \]
There are \( 6 \) such pairs.
4. **Calculate the number of reflexive relations**: Each of the remaining \( 6 \) pairs can either be included or excluded independently, which gives us:
\[ 2^6 = 64 \]

### Final Answer

- The total number of relations that can be defined in the set \( A \) is **512**.
- The number of reflexive relations is **64**.