Question

Write the folllowing sets in set-builder (Rule Method) form:
$B_3=\{\frac{1}{3},\frac{3}{5},\frac{5}{7},\frac{7}{9},\frac{9}{11}.....\}$

Answer 1

Step: Express the given set in Set builder (Rule Method) form:

In set builder form, the actual elements of the set are not written, but a statement or a formula or a rule is written in briefest possible way to represent the elements of the set.

Given:
$B_3=\{\frac{1}{3},\frac{3}{5},\frac{5}{7},\frac{7}{9},\frac{9}{11}.....\}$

As,

$3=1+2$
$5=3+2$
$7=5+2$
$9=7+2$
$11=9+2$
We can see that the denominator of every fraction (element of $B_3$) can be obtained by adding 2 to the numerator of same fraction.

So, elements of $B_3$ can be written as $\frac{n}{n+2}$, where n is an odd natural number

Hence, set builder form of $B_3$ is
$B_3=\{x:x=\frac{n}{n+2}$, where n is odd natural number}.