Question

Let $A=\{1,2,3\}$ and $B=\{1,2,3,4\}$ and let $R:A\rightarrow B$ be a relation. If $R=\{(1,2), (2,3), (3,4)\}$, then $R\circ R$ is

• A. \{(1,2), (2, 3), (3, 4)\}
• B. \{(1,2), (2, 1), (3, 4)\}
• C. \{(1,3), (2, 4)\}
• D. \{(1, 1), (2, 3), (3, 2)\}
• E. \{(1,2), (2, 1), (3, 3)\}

Hint:

Logic Tip: When calculating $R \circ R$, simply trace the path of the coordinates. If $A \rightarrow B$ and $B \rightarrow C$, then the composed path is $A \rightarrow C$. Make sure the intermediate value $B$ exactly matches!

Answer 1

The Correct Option is C

Concept:
The composition of two relations $R$ and $S$, denoted as $S \circ R$, is defined as: $$S \circ R = \{(a, c) \mid \exists b \text{ such that } (a, b) \in R \text{ and } (b, c) \in S\}$$ In this case, we are finding $R \circ R$, which means we apply the relation $R$ to itself.
Step 1: Evaluate the first pair in R.
The first element in $R$ is $(1, 2)$. This means $1$ maps to $2$. Now we look for a pair in $R$ that starts with $2$. We find $(2, 3)$. Since $(1, 2) \in R$ and $(2, 3) \in R$, their composition connects $1$ directly to $3$. $$(1, 3) \in R \circ R$$
Step 2: Evaluate the second pair in R.
The second element in $R$ is $(2, 3)$. This means $2$ maps to $3$. Now we look for a pair in $R$ that starts with $3$. We find $(3, 4)$. Since $(2, 3) \in R$ and $(3, 4) \in R$, their composition connects $2$ directly to $4$. $$(2, 4) \in R \circ R$$
Step 3: Evaluate the third pair in R.
The third element in $R$ is $(3, 4)$. This means $3$ maps to $4$. Now we look for a pair in $R$ that starts with $4$. There are no pairs in $R$ starting with $4$. Thus, this pair yields no new composite elements.
Step 4: Combine the resulting pairs.
The final composite relation $R \circ R$ contains the pairs generated in the steps above: $$R \circ R = \{(1, 3), (2, 4)\}$$