Question

Let we have two sets as \[A = \left\{ {x,y,z} \right\}\] and \[B = \left\{ {a,b,c,d} \right\}.\] which one of the following is not a relation from \[A\] to \[B\]
1). \[\left\{ {\left( {x,a} \right),\left( {x,c} \right)} \right\}\]
2). \[\left\{ {\left( {y,c} \right),\left( {y,d} \right)} \right\}\]
3). \[\left\{ {\left( {z,a} \right),\left( {z,d} \right)} \right\}\]
4). \[\left\{ {\left( {z,b} \right),\left( {y,b} \right),\left( {a,d} \right)} \right\}\]
5). \[\left\{ {\left( {x,c} \right)} \right\}\]

Answer 1

Hint: We need to find which one of the following is not a relation from \[A\] to \[B\]. We solve this question by using the concept of relations between two sets . We find the Cartesian product between A and B to find the relation set between the two sets. From the relation set we can conclude that which element of the relation set contains the element as stated in the option.

Complete step-by-step solution:
Given :
\[A = \left\{ {x,y,z} \right\}\] and \[B = \left\{ {a,b,c,d} \right\}\]
We have to find the relation from \[\;A\] to \[B\] , so we have to find the cartesian product of the sets in the manner of \[A \times B\] .
Now , for the set of elements of \[A \times B\] .
We multiply the elements of the first set with each element of the second set .
Let us consider if there are two sets $P$ and $Q$ such that \[P = \left\{ {1,2} \right\}\] and \[Q = \left\{ {a,b} \right\}\] , then the relation between $P$ and $Q$ from $P$ to $Q$ is given as :
\[P \times Q = \left\{ {\left( {p,q} \right):p \in P,q \in Q} \right\}\]
I.e. the relation from $P$ to $Q$ would be ,
\[P \times Q = \left\{ {\left( {1,a} \right),\left( {1,b} \right),\left( {2,a} \right),\left( {2,b} \right)} \right\}\]
The relation from $P$ to $Q$ does not co nation any element which has the element of set $Q$ first in the relation set .
i.e. the relation won’t have the elements like \[\left( {b,1} \right)\] or \[\left( {a,1} \right)\] etc .
So , using this concept we define the relation from $A$ to $B$ as given , \[\begin{array}{*{20}{l}}
{A \times B = \left\{ {\left( {x,a} \right),\left( {x,b} \right),\left( {x,c} \right),\left( {x,d} \right),\left( {y,a} \right),\left( {y,b} \right),\left( {y,c} \right),\left( {y,d} \right),\left( {z,a} \right),\left( {z,b} \right),\left( {z,c} \right),\left( {z,d} \right)} \right\}}
\end{array}\]
So the set of relations from $A$ to $B$ have these elements in it . All other elements than these stated above don’t be defined in the relation from $A$ to $B$ .
Thus from the given options the element \[\left\{ {\left( {a,d} \right)} \right\}\] does not belong to the set of elements of relation \[A \times B\] .
Hence , we conclude that the option \[\left( 4 \right)\] is not a relation from $A$ to $B$ .

Note: If either $P$ or $Q$ given set is an empty set , then \[P \times Q\] will also be an empty set .
In general \[A \times B \ne B \times A\]
A relation $R$ from a set $A$ to a set $B$ is a subset of the Cartesian product \[A \times B\] obtained by describing a relationship between the first element $x$ and the second element $y$ of the ordered pairs in \[A \times B\] .