Show that the number of equivalence relation in the set {1,2,3} containing (1,2) and (2,1) is two.
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Step-by-Step Solution
Step 1: Understand Equivalence Relations
An equivalence relation R on a set A must satisfy three properties: reflexivity, symmetry, and transitivity. The given set is A={1,2,3}. We are looking for equivalence relations that contain the ordered pairs (1,2) and (2,1).
Step 2: Apply Reflexivity
For any equivalence relation R on A, it must be reflexive. This means that for every element a∈A, the pair (a,a) must be in R. So, (1,1), (2,2), and (3,3) must be part of any equivalence relation on A.
Step 3: Incorporate Given Pairs and Symmetry
We are given that (1,2) and (2,1) are in the relation R. This already satisfies the symmetry property for these two elements. We combine these with the reflexive pairs to form a base for our relations.
Step 4: Consider Transitivity and Possible Partitions
Now we consider transitivity. Since (1,2)∈R and (2,1)∈R, transitivity implies that if (1,2)∈R and (2,x)∈R, then (1,x)∈R. Similarly for other combinations. This means that 1 and 2 must be in the same equivalence class. We have two possibilities for partitions of A that include 1 and 2 in the same class: \\ 1. The partition is {{1,2},{3}}. This corresponds to the equivalence relation R1={(1,1),(2,2),(3,3),(1,2),(2,1)}. \\ 2. The partition is {{1,2,3}}. This means all elements are related to each other, forming the universal relation R2=A×A={(1,1),(2,2),(3,3),(1,2),(2,1),(1,3),(3,1),(2,3),(3,2)}. Both R1 and R2 contain (1,2) and (2,1) and satisfy all properties of an equivalence relation.
Step 5: Conclusion
We have found two distinct equivalence relations on the set A={1,2,3} that contain the pairs (1,2) and (2,1). These are R1={(1,1),(2,2),(3,3),(1,2),(2,1)} and R2={(1,1),(2,2),(3,3),(1,2),(2,1),(1,3),(3,1),(2,3),(3,2)}.