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Step-by-Step Solution
Step 1: Understand the Initial Relation
The given relation is
$R = \{(a,b), (b,c)\}$
on the set $\{a, b, c\}$. We want to make $R$ both symmetric and transitive.
Step 2: Enforce Symmetry
A relation $R$ is symmetric if whenever $(x, y) \in R$, then $(y, x)$ is also in $R$. Currently, we have:
$(a, b) \in R$ but $(b,a)$ is missing.
$(b, c) \in R$ but $(c,b)$ is missing.
So, to satisfy symmetry, add these pairs:
$(b,a)$ and $(c,b)$.
Step 3: Check Transitivity and Add Missing Pairs
A relation $R$ is transitive if whenever $(x,y) \in R$ and $(y,z) \in R$, then $(x,z) \in R$ must also be in $R$. Let us check transitivity with the current pairs.
From $(a,b)$ and $(b,c)$, we need $(a,c)$ for transitivity.
Hence, add $(a,c)$. Then, by symmetry, we must also add $(c,a)$ because the relation must remain symmetric.
Step 4: Verify New Pairs for Transitivity Again
Now $R$ contains:
Original pairs: $(a,b), (b,c)$
Symmetry pairs: $(b,a), (c,b)$
New transitive pair: $(a,c)$ and its symmetric pair $(c,a)$
Check further transitivity among all these pairs:
$(a,b)$ and $(b,a)$ imply $(a,a)$ must be in $R$.
$(b,c)$ and $(c,b)$ imply $(b,b)$ must be in $R$.
$(a,c)$ and $(c,a)$ imply $(a,a)$ (already identified) and similarly $(c,c)$ if $(c,b)$ and $(b,c)$ combine in other ways.
So we see we must also add $(a,a), (b,b), (c,c)$ to ensure transitivity continues to hold for all possible pair combinations.
Step 5: Count the Number of Added Elements
We began with $2$ elements: $(a,b)$ and $(b,c)$.
We added $2$ elements for symmetry: $(b,a), (c,b)$.
We added $1$ element for transitivity: $(a,c)$, and $1$ more $(c,a)$ for symmetry of that new pair.
Finally, we added $3$ diagonal elements to maintain transitivity: $(a,a), (b,b), (c,c)$.
Thus, the total number of new elements added is
$2 + 1 + 1 + 3 = 7$.
Answer
The minimum number of elements that must be added to make the relation symmetric and transitive is 7.
