Question

Let $A=\left\{x,y,z\right\}$ and $B=\left\{a,b,c,d\right\}$. which one of the following is not a relation from $A$to $B$

• A. $\left\{\left(x,a\right),\left(x,c\right)\right\}$
• B. $\left\{\left(y,c\right),\left(y,d\right)\right\}$
• C. $\left\{\left(z,a\right),\left(z,d\right)\right\}$
• D. $\left\{\left(z,b\right),\left(y,b\right),\left(a,d\right)\right\}$
• E. $\left\{\left(x,c\right)\right\}$

Answer 1

The correct option is D

$\left\{\left(z,b\right),\left(y,b\right),\left(a,d\right)\right\}$

Explanation for the correct option:

Step 1: Determine the Cartesian Product.

We have,

$A=\left\{x,y,z\right\}$

$B=\left\{a,b,c,d\right\}$

From the above sets, the following Cartesian relation can be obtained :

$$\begin{gathered} \left\{\left(x,a\right),\left(x,b\right),\left(x,c\right),\left(x,d\right) \\ \left(y,a\right),\left(y,b\right),\left(y,c\right),\left(y,d\right) \\ \left(z,a\right),\left(z,b\right),\left(z,c\right),\left(z,c\right)\right\} \end{gathered}$$

Option D :

$\left\{\left(z,b\right),\left(y,b\right),\left(a,d\right)\right\}$, does not exits in the given cartesian product, hence can be the correct option.

Step 2: Determine the incorrect option.

Option A :

$\left\{\left(x,a\right),\left(x,c\right)\right\}$, exits in the given cartesian product, hence cannot be the correct option.

Option B :

$\left\{\left(y,c\right),\left(y,d\right)\right\}$, exits in the given cartesian product, hence cannot be the correct option.

Option C :

$\left\{\left(z,a\right),\left(z,d\right)\right\}$, exits in the given cartesian product, hence cannot be the correct option.

Option E :

$\left\{\left(x,c\right)\right\}$, exits in the given cartesian product, hence cannot be the correct option.

Hence, the correct answer is Option D.