Question

What is \(R^3\) of \[ R=\{(1,1),(2,1),(3,2),(4,3)\}? \]

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

Hint:

To compute \(R^n\), repeatedly follow the arrows of the relation \(n\) times. A relation can often be interpreted as a directed graph, making powers easier to calculate.

Answer 1

The Correct Option is C

Concept: For a relation \(R\), \[ R^2=R\circ R, \] and \[ R^3=R^2\circ R. \] The composition of relations is obtained by linking ordered pairs through common intermediate elements.

Step 1: Write the given relation.
\[ R=\{(1,1),(2,1),(3,2),(4,3)\}. \] Observe the mappings: \[ 1\to1, \] \[ 2\to1, \] \[ 3\to2, \] \[ 4\to3. \]

Step 2: Find \(R^2\).
Following each element twice: \[ 1\to1\to1 \] gives \[ (1,1). \] \[ 2\to1\to1 \] gives \[ (2,1). \] \[ 3\to2\to1 \] gives \[ (3,1). \] \[ 4\to3\to2 \] gives \[ (4,2). \] Therefore, \[ R^2=\{(1,1),(2,1),(3,1),(4,2)\}. \]

Step 3: Find \(R^3\).
Apply \(R\) once more: \[ 1\to1\to1\to1 \] gives \[ (1,1). \] \[ 2\to1\to1\to1 \] gives \[ (2,1). \] \[ 3\to2\to1\to1 \] gives \[ (3,1). \] \[ 4\to3\to2\to1 \] gives \[ (4,1). \] Hence, \[ R^3= \{(1,1),(2,1),(3,1),(4,1)\}. \] \[ \boxed{\{(1,1),(2,1),(3,1),(4,1)\}} \] Thus option (C) is correct.