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Step 1: Identify the Requirements of an Equivalence Relation
An equivalence relation on a set must satisfy three properties:
Reflexivity: For every a in A , (a,a) must be in the relation.
Symmetry: If (a,b) is in the relation, then (b,a) must also be in the relation.
Transitivity: If (a,b) and (b,c) are in the relation, then (a,c) must be in the relation.
Step 2: Incorporate the Given Relation R into S
We have A = \{1, 2, 3, 4\} and R = \{(1,2), (2,3), (1,4)\} . The relation S must be an equivalence relation that contains R , i.e., R \subseteq S .
Step 3: Ensure Reflexivity
Equivalence relations require that each element relates to itself. Hence, we must include:
(1,1), (2,2), (3,3), (4,4).
Step 4: Ensure Symmetry
Since R contains (1,2) , (2,3) , and (1,4) , to make S symmetric, we must add the reverse pairs as well:
(2,1), (3,2), (4,1).
Step 5: Ensure Transitivity
Transitivity requires that whenever (a,b) and (b,c) are in S , (a,c) must also be in S . Observe the implications:
From (1,2) and (2,3) , you must include (1,3) . By symmetry, add (3,1) .
Because (1,4) is also in the relation, transitivity between (1,2) , (2,3) , and (1,4) eventually forces 1, 2, 3, and 4 to be in the same equivalence class.
If all elements end up in a single equivalence class, then every possible pair (a,b) for a,b \in \{1,2,3,4\} must be included. That is the smallest equivalence relation containing R because any partial connection between some elements quickly propagates to connect them all.
Step 6: Count the Number of Ordered Pairs
In a set of 4 elements, the total number of ordered pairs in an equivalence relation that forms one single equivalence class is 4 \times 4 = 16. Thus, the minimum size of S is:
n = 16.
