Let A = {a, b, c} andR = {(a, b), (b, c)}. The minimum number of ordered pairs that must be added to R to make it an equivalence relation is

Options

A. 5
B. 6
C. 7
D. 8

Answer

To determine the minimum number of ordered pairs that must be added to the relation \( R \) to make it an equivalence relation, we need to ensure that the relation satisfies three properties: reflexivity, symmetry, and transitivity.

### Step 1: Check for Reflexivity
A relation is reflexive if every element in the set relates to itself. For the set \( A = \{a, b, c\} \), we need the following pairs to be present in \( R \):
- \( (a, a) \)
- \( (b, b) \)
- \( (c, c) \)

Currently, \( R = \{(a, b), (b, c)\} \) does not contain any of these pairs. Therefore, we need to add:
- \( (a, a) \)
- \( (b, b) \)
- \( (c, c) \)

**Pairs added for reflexivity:** 3 pairs

### Step 2: Check for Symmetry
A relation is symmetric if for every pair \( (x, y) \) in \( R \), the pair \( (y, x) \) is also in \( R \).

From the current relation \( R \):
- We have \( (a, b) \), so we need to add \( (b, a) \).
- We have \( (b, c) \), so we need to add \( (c, b) \).

**Pairs added for symmetry:** 2 pairs

### Step 3: Check for Transitivity
A relation is transitive if whenever \( (x, y) \) and \( (y, z) \) are in \( R \), then \( (x, z) \) must also be in \( R \).

From the pairs we have:
- We have \( (a, b) \) and \( (b, c) \), so we need to add \( (a, c) \).

Now, after adding \( (a, c) \):
- We have \( (a, c) \) and \( (c, b) \), so we need to add \( (c, a) \) to maintain symmetry.

**Pairs added for transitivity:** 2 pairs

### Summary of Pairs Added
- Reflexivity: 3 pairs \( (a, a), (b, b), (c, c) \)
- Symmetry: 2 pairs \( (b, a), (c, b) \)
- Transitivity: 1 pair \( (a, c) \) and then \( (c, a) \)

### Total Pairs Added
Adding all the pairs together:
- Reflexive pairs: 3
- Symmetric pairs: 2
- Transitive pairs: 2

Total = \( 3 + 2 + 2 = 7 \)

Thus, the minimum number of ordered pairs that must be added to \( R \) to make it an equivalence relation is **7**.