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Step-by-Step Solution
Step 1: Recognize the Properties of an Equivalence Relation
An equivalence relation on a set must satisfy:
Reflexivity: Every element is related to itself.
Symmetry: If $a$ is related to $b$, then $b$ is also related to $a$.
Transitivity: If $a$ is related to $b$, and $b$ is related to $c$, then $a$ is also related to $c$.
We want the smallest such relation $R$ on the set $\{1,2,3,4\}$ containing the pairs $(1,2)$ and $(1,3)$.
Step 2: Include All Reflexive Pairs
By reflexivity, each element relates to itself, so we must include:
$ (1,1), (2,2), (3,3), (4,4) $
Step 3: Enforce Symmetry for the Given Pairs
Since $(1,2)$ and $(1,3)$ are in $R$, by symmetry we need:
$ (2,1) $ (because $(1,2)$ is in $R$)
$ (3,1) $ (because $(1,3)$ is in $R$)
Step 4: Apply Transitivity
We see that $1$ is related to $2$ and $1$ is related to $3$. By transitivity, $2$ must be related to $3$ and $3$ must be related to $2$. Thus we add:
$ (2,3), (3,2) $
Step 5: Form the Equivalence Class for 1, 2, and 3
Because $1$, $2$, and $3$ must all be in the same equivalence class, the relation among them includes every possible pairing (reflexive, symmetric, transitive):
$ (1,1), (2,2), (3,3) $
$ (1,2), (2,1), (1,3), (3,1), (2,3), (3,2) $
There are $9$ pairs among $1$, $2$, and $3$.
Step 6: Account for Element 4
The element $4$ has no given requirement to be in the same class as $1$, $2$, or $3$. Hence, it stands in its own equivalence class, which contains only:
$ (4,4) $
Step 7: Count the Total Number of Pairs
Pairs among $1$, $2$, and $3$: $9$
Pair for $4$: $1$ (namely $ (4,4) $)
Total pairs in $R = 9 + 1 = 10$.
Final Answer
The number of elements in the smallest equivalence relation containing the given pairs is 10.
