Question

Let set $A = \\{ 1,2,3,......,14\\} .$ Define a relation R from A to A by R = \\{ (x,y):3x - y = 0 where x.y \in A\\} . Write down its domain, co-domain and range.

Answer 1

According to given in the question we have to determine domain, co-domain and range for let $A = \\{ 1,2,3,......,14\\}.$ Define a relation R from A to A by R = \\{ (x,y):3x - y = 0 where x.y \in A\\} . So, first of all we have to rearrange the terms of the expression as given in the relation and now, we have to understand about domain, co-domain and range which is as explained below.
Domain: The domain of a function is the complete set of possible values of the independent variables.
Now, we have to understand about the co-domain which is as explained below:
Co-domain: The codomain of a function is the set of its possible outputs and in the function machine metaphor, the codomain is the set of objects that might possibly come out of the machine.
Now, we have to understand about the range which is as explained below:
Range: The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually) after we have substituted the domain.
Now, we have to substitute the range which is $A = \\{ 1,2,3,......,14\\}.$ as given in the question to determine the domain, co-domain and range.

Complete step-by-step answer:
Step 1: First of all we have to rearrange the terms of the expression as given in the relation as mentioned in the solution hint.
$\Rightarrow R = \\{ (x,y):3x = y;x.y \in A\\} \\\ \Rightarrow R = \\{ (x,y):y = 3x;x.y \in A\\} \\\$
Step 2: Now, we have to substitute the range in the relation which is as obtained in the solution step 1. Hence,
$\Rightarrow R = \\{ (1,3),(2,6),(3,9),(4,12)\\}$
Step 3: Now, we have to determine the domain about which we have already understood the solution hint and the domain of R is the set of all the first elements of the ordered pair in the relation.
$\Rightarrow$Domain of $R = \\{ 1,2,3,4\\}$
Step 4: Now, we have to determine the co-domain about which we have already understood the solution hint and the co-domain of the relation.
$\Rightarrow$Co-domain of $R = A = \\{ 1,2,3,..........,14\\}$
Step 5: Now, we have to determine the co-domain about which we have already understood in the solution hint the range of R is the set of all the second elements of the ordered pairs in the relation.
$\Rightarrow$Range of $R = \\{ 3,6,9,12\\}$

Hence, we have obtained domain which is $R = \\{ 1,2,3,4\\}$, co-domain which is $R = A = \\{ 1,2,3,..........,14\\}$ and range which is $R = \\{ 3,6,9,12\\}$.

Note:
The codomain and range are both on the output side, but are subtly different and co-domain is the set of values that could possibly come out; the co-domain is actually part of the definition of the function.
The range is basically the difference between the largest and the smallest numbers and the midrange is the average of the largest and smaller number.