Q.Write the following sets in the roster form:

Here are the sets in roster form:

(i) $A = \{x : x \in \mathbb{R},\ 2x + 11 = 15\}$

The key idea is to solve the linear equation to find the elements of the set.

1. The condition for $x \in A$ is $2x + 11 = 15$.
2. Subtracting 11 from both sides gives $2x = 4$.
3. Dividing by 2 yields $x = 2$.
4. Since $2$ is a real number, it is the only element satisfying the condition.

(ii) $B = \{x \mid x^2 = x,\ x \in \mathbb{R}\}$

The key idea is to solve the quadratic equation to find the elements of the set.

1. The condition for $x \in B$ is $x^2 = x$.
2. Rearranging the equation gives $x^2 - x = 0$.
3. Factoring out $x$ yields $x(x - 1) = 0$.
4. This implies $x = 0$ or $x - 1 = 0$, so $x = 0$ or $x = 1$. Both $0$ and $1$ are real numbers.

(iii) $C = \{x \mid x \text{ is a positive factor of a prime number } p\}$

The key idea is to understand the definition of a prime number and its factors.

1. A prime number $p$ is a natural number greater than 1.
2. By definition, a prime number has exactly two distinct positive factors.
3. These factors are always 1 and the prime number itself.
4. Therefore, the positive factors of any prime number $p$ are $1$ and $p$.

When a set is given in set-builder notation, like $S = \{x : \text{condition}(x)\}$, it describes the properties that elements $x$ must possess to be part of the set. To write such a set in roster form, we need to find all specific values of $x$ that satisfy these conditions and then list them explicitly within curly braces. This process involves solving equations or understanding definitions related to the conditions.

Let's break down each set.

(i) $A = \{x : x \in \mathbb{R},\ 2x + 11 = 15\}$

1. Understand the condition: The set $A$ consists of all real numbers $x$ such that $2x + 11 = 15$. The core task is to solve this linear equation for $x$.

2. Solve the equation:

   We have the equation:

   $2x + 11 = 15$

   Subtract $11$ from both sides:

   $2x = 15 - 11$

   $2x = 4$

   Divide by $2$:

   $x = \frac{4}{2}$

   $x = 2$

3. Check the domain: The condition states $x \in \mathbb{R}$ ( $x$ is a real number). Our solution $x=2$ is indeed a real number.

4. Write in roster form: Since $x=2$ is the only value that satisfies the condition, the set $A$ contains only this element.

   $A = \{2\}$

(ii) $B = \{x \mid x^2 = x,\ x \in \mathbb{R}\}$

1. Understand the condition: The set $B$ consists of all real numbers $x$ such that $x^2 = x$. We need to solve this quadratic equation.

2. Solve the equation:

   The equation is:

   $x^2 = x$

   To solve a quadratic equation, it's best to bring all terms to one side and set the expression to zero:

   $x^2 - x = 0$

   Now, factor out the common term $x$:

   $x(x - 1) = 0$

   For the product of two terms to be zero, at least one of the terms must be zero. So, we have two possibilities:

   $x = 0 \quad \text{or} \quad x - 1 = 0$

   Solving the second part:

   $x - 1 = 0 \implies x = 1$

   So, the solutions are $x=0$ and $x=1$.

   Watch out

   A common mistake here is to divide both sides of $x^2 = x$ by $x$, leading to $x=1$. This loses the solution $x=0$. Always move all terms to one side and factor when solving equations involving variables in denominators or when a variable could be zero.

3. Check the domain: The condition states $x \in \mathbb{R}$. Both $0$ and $1$ are real numbers.

4. Write in roster form: The set $B$ contains the elements $0$ and $1$.

   $B = \{0, 1\}$

(iii) $C = \{x \mid x \text{ is a positive factor of a prime number } p\}$

1. Understand the condition: The set $C$ consists of all $x$ such that $x$ is a positive factor of a prime number $p$. This means we need to understand what a prime number is and what its factors are.

2. Define a prime number:

   Important

   A prime number is a natural number greater than 1 that has exactly two distinct positive divisors: 1 and itself.

   Examples: $2, 3, 5, 7, 11, \dots$

3. Identify factors of a prime number:

   Let $p$ be any prime number. By its definition, the only positive numbers that divide $p$ evenly are $1$ and $p$.

   For example:

   • If $p=2$, its positive factors are $1, 2$.
   • If $p=3$, its positive factors are $1, 3$.
   • If $p=7$, its positive factors are $1, 7$.

   In general, for any prime number $p$, its positive factors are always $1$ and $p$.

4. Write in roster form: The elements of set $C$ are these positive factors.

   $C = \{1, p\}$