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Q1(ii):
Which of the following are examples of the null set (ii) Set of even prime numbers
Solution :
Definition of Null Set (Empty Set)
A set is defined as a null set (or empty set), denoted by $\phi$ or $\{\}$, if it contains no elements [Per the Axiom of Extensionality].
Step 1: Identifying the Mathematical Constraints
The given condition defines the set $A$ as: $A = \{x : x \text{ is an even prime number}\}$
To evaluate if this is a null set, we must identify the set of prime numbers and apply the parity filter.
Step 2: Analysis of Prime Numbers
A prime number $p$ is a natural number greater than $1$ that has exactly two distinct positive divisors: $1$ and $p$.
- The set of the first few prime numbers is $\{2, 3, 5, 7, 11, 13, \dots\}$.
- Parity check: A number is even if it is divisible by $2$.
- In the set of prime numbers, $2$ is the only even prime number because all other even numbers are divisible by $2$, and thus possess more than two divisors [By the Fundamental Theorem of Arithmetic].
Step 3: Verification of the Set Composition
Since the number $2$ satisfies both conditions (being prime and being even), the set $A$ can be explicitly listed using the Roster Method:
$A = \{2\}$
Step 4: Cardinality Comparison
The cardinality of set $A$, denoted as $n(A)$, is the number of elements in the set.
| Set | Cardinality $n(A)$ | Classification |
|---|---|---|
| Null Set ($\phi$) | $0$ | Empty |
| Set of even prime numbers | $1$ | Singleton Set |
Because $n(A) = 1 \neq 0$, the set is not empty.
Final Solution: The set of even prime numbers is not a null set, as it contains exactly one element: $\{2\}$. Therefore, it is a singleton set.
More Questions from Class 11 Mathematics Sets EXERCISE 1.2
- Q1(i): Which of the following are examples of the null set (i) Set of odd natural numbers divisible by 2
- Q1(iii): Which of the following are examples of the null set (iii) \{ $x$ : $x$ is a natural numbers, $x < 5$ and $x > 7$ \}
- Q1(iv): Which of the following are examples of the null set (iv) \{ $y$ : $y$ is a point common to any two parallel lines\}
- Q2(i): Which of the following sets are finite or infinite (i) The set of months of a year
- Q2(ii): Which of the following sets are finite or infinite (ii) \{1, 2, 3, . . .\}
- Q2(iii): Which of the following sets are finite or infinite (iii) \{1, 2, 3, . . .99, 100\}
- Q2(iv): Which of the following sets are finite or infinite (iv) The set of positive integers greater than 100
- Q2(v): Which of the following sets are finite or infinite (v) The set of prime numbers less than 99
- Q3(i): State whether each of the following set is finite or infinite: (i) The set of lines which are parallel to the x-axis
- Q3(ii): State whether each of the following set is finite or infinite: (ii) The set of letters in the English alphabet
- Q3(iii): State whether each of the following set is finite or infinite: (iii) The set of numbers which are multiple of 5
- Q3(iv): State whether each of the following set is finite or infinite: (iv) The set of animals living on the earth
- Q3(v): State whether each of the following set is finite or infinite: (v) The set of circles passing through the origin (0,0)
- Q4(i): In the following, state whether $A = B$ or not: (i) $A = \{ a, b, c, d \}$ $B = \{ d, c, b, a \}$
- Q4(ii): In the following, state whether $A = B$ or not: (ii) $A = \{ 4, 8, 12, 16 \}$ $B = \{ 8, 4, 16, 18\}$
- Q4(iii): In the following, state whether $A = B$ or not: (iii) $A = \{2, 4, 6, 8, 10\}$ $B = \{ x : x$ is positive even integer and $x \le 10\}$
- Q4(iv): In the following, state whether $A = B$ or not: (iv) $A = \{ x : x$ is a multiple of 10\}, $B = \{ 10, 15, 20, 25, 30, . . . \}$
- Q5(i): Are the following pair of sets equal ? Give reasons. (i) $A = \{2, 3\}$, $B = \{x : x$ is solution of $x^2 + 5x + 6 = 0\}$
- Q5(ii): Are the following pair of sets equal ? Give reasons. (ii) $A = \{ x : x$ is a letter in the word FOLLOW\} $B = \{ y : y$ is a letter in the word WOLF\}
- Q6: From the sets given below, select equal sets : $A = \{ 2, 4, 8, 12\}$, $B = \{ 1, 2, 3, 4\}$, $C = \{ 4, 8, 12, 14\}$, $D = \{ 3, 1, 4, 2\}$ $E = \{–1, 1\}$, $F = \{ 0, a\}$, $G = \{1, –1\}$, $H = \{ 0, 1\}$
CBSE Solutions for Class 11 Mathematics Sets
Chapters in CBSE - Class 11 Mathematics
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