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Step-by-Step Solution
Step 1: Understand the Problem
We are given a relation
R = \{(a, b), (b, c), (b, d)\}
on the set
\{a, b, c, d\}.
We must add the minimum number of ordered pairs to R so that it becomes an equivalence relation. An equivalence relation must be:
Reflexive: (x, x) is in the relation for every element x in the set.
Symmetric: If (x, y) is in the relation, then (y, x) must also be in the relation.
Transitive: If (x, y) and (y, z) are in the relation, then (x, z) must also be in the relation.
Step 2: Make the Relation Reflexive
For reflexivity on the set \{a, b, c, d\}, we must add:
(a, a)
(b, b)
(c, c)
(d, d)
So we add 4 ordered pairs to ensure reflexivity.
Step 3: Make the Relation Symmetric
Currently, R has the pairs (a, b), (b, c), (b, d). For each of these, we must include their “reverse” counterparts:
(a, b) exists, so add (b, a) .
(b, c) exists, so add (c, b) .
(b, d) exists, so add (d, b) .
Hence, we add 3 more pairs for symmetry so far.
Step 4: Make the Relation Transitive and Maintain Symmetry
Check transitivity among the existing and newly added pairs. For transitivity, if (x, y) and (y, z) are in the relation, then (x, z) must be added (and its symmetric pair as well). Examining all possibilities yields the need to add:
From (a, b) and (b, c) , add (a, c) .
From (a, b) and (b, d) , add (a, d) .
From (c, b) and (b, a) , add (c, a) .
From (c, b) and (b, d) , add (c, d) .
From (d, b) and (b, a) , add (d, a) .
From (d, b) and (b, c) , add (d, c) .
Each new transitive pair also requires its symmetric pair if not already added:
(a, c) and (c, a)
(a, d) and (d, a)
(c, d) and (d, c)
These 6 pairs (accounting directly for their symmetric counterparts) are necessary for the relation to be transitive and remain symmetric.
Step 5: Count the Total Number of Added Pairs
We have added:
4 pairs for reflexivity: (a,a), (b,b), (c,c), (d,d)
3 pairs for initial symmetry: (b,a), (c,b), (d,b)
6 pairs for transitivity (along with their symmetric versions):
(a,c), (c,a), (a,d), (d,a), (c,d), (d,c)
Total new pairs added = 4 + 3 + 6 = 13.
Final Answer
Therefore, the minimum number of elements that must be added to R to make it an equivalence relation is
\boxed{13}.
