Question

The minimum number of elements that must be added to the relation $R=\{(a, b),(b, c),(b, d)\}$ on the set $\{a, b, c, d\}$ so that it is an equivalence relation, is _______

Answer 1

Correct Answer: 13

Step 1: Ensure Reflexivity 

For reflexivity, each element in the set \( \{ a, b, c, d \} \) must relate to itself. Thus, the following pairs need to be added:

\[ (a, a), (b, b), (c, c), (d, d). \] 

Step 2: Ensure Symmetry

For symmetry, if \( (x, y) \in R \), then \( (y, x) \) must also be in \( R \). The given relation is:

\[ R = \{ (a, b), (b, c), (b, d) \}. \]

To ensure symmetry, the following pairs need to be added:

\[ (b, a), (c, b), (d, b). \] 

Step 3: Ensure Transitivity

For transitivity, if \( (x, y) \in R \) and \( (y, z) \in R \), then \( (x, z) \) must also be in \( R \). Adding the necessary pairs for transitivity gives:

\[ (a, c), (a, d), (c, a), (d, a), (c, d), (d, c). \] 

Step 4: Total pairs to be added

The total number of pairs to be added is:

\[ 4 \text{ (reflexive)} + 3 \text{ (symmetric)} + 6 \text{ (transitive)} = 13. \]