Define a relation on R

Question

Given the following proof:

Let A = {2,3,4,5} and B = {9,16,20}. Define a relation R from A to B by aRb if and only if (i) a|b and (b\a) - 1 is a prime number.

How would I create said relation on R and determine if it is a function or not?

Answer 1

You first check each element of $A$ and find out which elements of $B$ it is related to. In order for this to define a function, it must be well defined. That means that every element in $A$ must be related to something in $B$, and can't be related to more than one thing.

Then you have something mapping every element in $A$ to something in $B$, and you have a well defined function. If any of the above conditions break, you don't.

Answer 2

If I am interpreting correctly,

$a$ are the elements of $ A$ and $b$ are elements of $B$

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For (i) $ a|b $ Which implies " $a$ divides $b$ exactly ",

The required relation in roster form in this case is,

$ R = { (2,16) , (2,20) , (4,16) , (4,20) , (5,20) } $

For (ii) $ \frac{b}{a} - 1 $ is prime,

$ R_1 = {(3,9) , (2,16), (4,16) , (5,20)} $

Since, in $R$ , the element "$2$" of set $A$ has two images in set $B$ , so this relation isn't a function.

While in $R_1$ each element of $A$ has unique image in $B$, So this relation is a function.

Now if we combine both $R$ and $R_1$ we get, $ R_{final} = {(2,16),(4,16),(5,20)} $

In this, each element doesn't have an image on $B$. So it is NOT a function.