Write each of the sets given below in set – builder form: (i) C = {53, 59, 61, 67, 71, 73, 79}(ii) D = {– 1, 1}(iii) E = {14, 21, 28, 35, 42, …., 98}

Here is the solution to the question, written in detail:

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**Question:** Write each of the sets given below in set-builder form.

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### (i) $C=\{53,59,61,67,71,73,79\}$

**Solution:**

The given set $C$ contains prime numbers between 50 and 80.
In set-builder notation, $C$ can be expressed as:

$$C=\{x|x\text{ is a prime number and }50<x<80\}$$

This means $x$ belongs to the set of all prime numbers, and $x$ satisfies the condition $50<x<80$.

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### (ii) $D=\{-1,1\}$

**Solution:**

The given set $D$ contains only two elements, $-1$ and $1$. These are the solutions to the equation $x^2=1$.
In set-builder notation, $D$ can be expressed as:

$$D=\{x|x^2=1\}$$

This means $x$ belongs to the set of all numbers such that $x^2=1$.

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### (iii) $E=\{14,21,28,35,42,\dots ,98\}$

**Solution:**

The given set $E$ is an arithmetic progression (AP) with the first term $a=14$ and the common difference $d=7$. The general term of an AP is:

$$T_n=a+(n-1)d$$

Substitute $a=14$ and $d=7$:

$$T_n=14+(n-1)\cdot 7=7n+7$$

For $T_n\le 98$:

$$7n+7\le 98⟹7n\le 91⟹n\le 13$$

Thus, $n$ ranges from $1$ to $13$.
In set-builder notation, $E$ can be expressed as:

$$E=\{x|x=7n,n\in \mathbb{N},2\le n\le 14\}$$

This means $x$ belongs to the set of all numbers that can be expressed as $7n$, where $n$ is a natural number between 2 and 14 (inclusive).

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**Final Answer:**
(i) $C=\{x|x\text{ is a prime number and }50<x<80\}$
(ii) $D=\{x|x^2=1\}$
(iii) $E=\{x|x=7n,n\in \mathbb{N},2\le n\le 14\}$