Question

Let $A = \{ x,y,z\} $ and $B = \{ a,b,c,d\} $. Which one of the following is not a relation from A to B.
$1)\{ (x,a),(x,c)\} $
$2)\{ (y,c),(y,d)\} $
$3)\{ (z,a),(z,d)\} $
$4)\{ (z,b),(y,b),(a,d)\} $
$5)\{ (x,c)\} $

Answer 1

Hint: First, we will need to know about the concept of relation.
A relation $M$ is the subset of the cartesian product of M and N, where M and N are considered as two non-empty sets. It is concluded by stating their relationship between the first and second elements of the ordered pair.
Complete step-by-step solution:
Since from the given that we have two sets $A = \{ x,y,z\} $ and $B = \{ a,b,c,d\} $ we need to find their relation from A to B.
First staring with the option $1)\{ (x,a),(x,c)\} $ which is a relation from A to B because it consists of the elements in the A at the first and also the elements in the B at the second terms
Hence the option $1)\{ (x,a),(x,c)\} $ is incorrect because they are asking the not relation A to B.
Now we will go with the option $2)\{ (y,c),(y,d)\} $ which is a relation from A to B because it consists of the elements in the A at the first and also the elements in the B at the second terms
Hence the option $2)\{ (y,c),(y,d)\} $ is incorrect because they are asking the not relation A to B.
Now we will go with the option $3)\{ (z,a),(z,d)\} $ which is a relation from A to B because it consists of the elements in the A at the first and also the elements in the B at the second terms
Hence the option $3)\{ (z,a),(z,d)\} $ is incorrect because they are asking the not relation A to B.
Now we will go with the option $5)\{ (x,c)\} $ which is a relation from A to B because it consists of the elements in the A at the first and also the elements in the B at the second terms
Hence the option $5)\{ (x,c)\} $ is incorrect because they are asking the not relation A to B.
Finally take the option $4)\{ (z,b),(y,b),(a,d)\} $ which is not the relation from A to B, because the set $(a,d)$ is only contained on the B. hence it is not the relation A to B as both the elements of the sets need to contain in the subset.
Therefore, the option $4)\{ (z,b),(y,b),(a,d)\} $ is correct, because they are asking the not relation A to B subsets.

Note:Since a function is known as the relation only if each element of non-empty set M, has only one range to a non-empty set N.
Since if the question is about finding the relation function from A to B, then options like $1,2,3,5$ are the correct options and thus the option $4$ is incorrect. We need to be careful at the given question because correctly checking whether they are asking correct statements is incorrect as the requirements.