Question

Express in roster form:
A = {x + 1: 4 x + 7 < 25, x $\in$N}

Answer 1

We will use the definition of roster form of the set that the way of describing the content of a set by listing the elements in a set of curly brackets separated by commas is called the roster form of the set.

Complete step by step answer:

We need to express it in terms of roster form of a set.
For this, we will use the definition of the roster form of the set and using it, we will determine the elements of the set in a roster form.
Definition of roster form:
We are given 4 x + 7 < 25
$\Rightarrow$4 x < 25 – 7
$\Rightarrow$4 x < 18
$\Rightarrow$x < $\dfrac{{18}}{4} = \dfrac{9}{2}$
$\Rightarrow$x < $\dfrac{9}{2}$
We have used the given inequality to solve for the value of x and it comes out to be $\dfrac{9}{2}$.i.e., x can only take values less than $\dfrac{9}{2}$= 4.5 (excluding 4.5)
Now, we are given the variable as x + 1: 4 x + 7 < 25, therefore, we can say
$\Rightarrow$x + 1 <$\dfrac{9}{2}$+ 1
$\Rightarrow$x + 1 < $\dfrac{{9 + 2}}{2} = \dfrac{{11}}{2}$
$\Rightarrow$x + 1 < $\dfrac{{11}}{2}$
Therefore, x + 1 can only be a natural number less than $\dfrac{{11}}{2}$= 5.5
For x $\in$N, we have values of the elements in set A as:
For x = 1, 2, 3, 4 < $\dfrac{9}{2}$
$\Rightarrow$x + 1 = 2, 3, 4, 5 < $\dfrac{{11}}{2}$
$\Rightarrow$A = {2, 3, 4, 5}
Therefore, the set in the roster form is A = {2, 3, 4, 5}.

Note: In this question, you may get confused while solving the inequalities for obtaining the restriction of values of x. you may get wrong later on because we need to find the set elements in terms of x + 1 instead of x. You should be careful while substituting the values of x as you may neglect the fact that x is a variable belonging in the set of natural numbers. So, you can only consider xN i.e., x{1, 2, 3, 4, 5, …}. A set can be represented in two ways: roster form and set – builder form. We have used the set – builder form of the set to convert it into the roster form of the set.