Question

Let \[A = \left\{ {1,2,3} \right\}\] and \[B = \left\{ {2,3,4} \right\},\] then which of the following relations is a function from A to B?
A) $ \{ (1,2),(2,3),(3,4),(2,2)\} $
B) $ \{ (1,2),(2,3),(1,3)\} $
C) $ \{ (1,3),(2,3),(3,3)\} $
D) $ \{ (1,1),(2,3),(3,4)\} $

Answer 1

Hint: To solve this problem, that is to find from the options which of the relations is a function from A to B, so for that we will first find the cartesian product of A and B, i.e., \[A \times B.\]Then afterwards we will check from the options, that which of the relations comes in that and hence with that we will get our required answer.

Complete step-by-step answer:
We have been given that \[A = \left\{ {1,2,3} \right\}\] and \[B = \left\{ {2,3,4} \right\},\] then we need to find out of four multiple choices relations that which of them is a function from A to B.
So, we have, \[A = \left\{ {1,2,3} \right\}\] and \[B = \left\{ {2,3,4} \right\}.\]
Then, \[A{\text{ }} \times {\text{ }}B{\text{ }} = {\text{ }}\left\{ {\left( {1,{\text{ }}2} \right),{\text{ }}\left( {1,{\text{ }}3} \right),{\text{ }}\left( {1,4} \right),{\text{ }}\left( {2,{\text{ }}2} \right),{\text{ }}\left( {2,{\text{ }}3} \right),{\text{ }}\left( {2,{\text{ }}4} \right),{\text{ }}\left( {3,{\text{ }}2} \right),{\text{ }}\left( {3,{\text{ }}3} \right),{\text{ }}\left( {3,{\text{ }}4} \right)} \right\}\]
Now let us check the options given to us one by one.
Option (A), Let \[{R_1} = \] $ \{ (1,2),(2,3),(3,4),(2,2)\} $
Here, \[\left( {1,2} \right) \in A{\text{ }} \times {\text{ }}B\]
\[
\left( {2,3} \right) \in A{\text{ }} \times {\text{ }}B \\
\left( {3,4} \right) \in A{\text{ }} \times {\text{ }}B \\
\left( {2,2} \right) \in A{\text{ }} \times {\text{ }}B \\
 \]
 $ {R_1} \subseteq{\text{ }} \times {\text{ }}B $
Hence, \[{R_1}\] is a relation which is a function from A to B.

Option (B), Let \[{R_2} = \] $ \{ (1,2),(2,3),(1,3)\} $
Here, \[\left( {1,2} \right) \in A{\text{ }} \times {\text{ }}B\]
\[
\left( {2,3} \right) \in A{\text{ }} \times {\text{ }}B \\
\left( {1,3} \right) \in A{\text{ }} \times {\text{ }}B \\
 \]
Hence, \[{R_2}\] is a relation which is a function from A to B.

Option (C), Let \[{R_3} = \] $ \{ (1,3),(2,3),(3,3)\} $
Here, \[\left( {1,3} \right) \in A{\text{ }} \times {\text{ }}B\]
\[
\left( {2,3} \right) \in A{\text{ }} \times {\text{ }}B \\
\left( {3,3} \right) \in A{\text{ }} \times {\text{ }}B \\
 \]
Hence, \[{R_3}\] is a relation which is a function from A to B.

Option (D), Let \[{R_4} = \] $ \{ (1,1),(2,3),(3,4)\} $
Here, \[\left( {1,1} \right) \notin A{\text{ }} \times {\text{ }}B\]
\[
\left( {2,3} \right) \in A{\text{ }} \times {\text{ }}B \\
\left( {3,4} \right) \in A{\text{ }} \times {\text{ }}B \\
 \]
Hence, \[{R_4}\] is not a relation which is a function from A to B.
So, \[{R_1},{R_2},{R_3}\] is a relation which is a function from A to B.
Thus, options A) $ \{ (1,2),(2,3),(3,4),(2,2)\} $ , (B) $ \{ (1,2),(2,3),(1,3)\} $ and (C) $ \{ (1,3),(2,3),(3,3)\} $ , are correct.

So, the correct answer is “Option A,B AND C”.

Note: In the solutions we have taken the cartesian product of two given sets A and B, denoted as\[A \times \;B,\]is the set of all possible ordered pairs where the elements of A are placed first and the elements of B are placed second. And on writing that in the set-builder notation, we get, \[A \times \;B\; = \{ (a,\;b):a \in A{\text{ }}and\;b\; \in \;B\} .\]