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Q1(iii):
Which of the following are examples of the null set (iii) \{ $x$ : $x$ is a natural numbers, $x < 5$ and $x > 7$ \}
Solution :
Step 1: Formal Definition of the Set
The given set is defined in set-builder notation as:
$A = \{x : x \in \mathbb{N}, x < 5 \text{ and } x > 7\}$
Where $\mathbb{N}$ denotes the set of all natural numbers $\{1, 2, 3, \dots\}$.
Step 2: Logical Decomposition of Conditions
The set $A$ is defined by the conjunction (logical "and") of two distinct inequalities acting upon the same variable $x$:
- Condition 1 ($P$): $x < 5$
- Condition 2 ($Q$): $x > 7$
For an element $x$ to be a member of set $A$, it must satisfy $P \land Q$ simultaneously [Per the Principle of Conjunction in Set Theory].
Step 3: Interval and Domain Analysis
We analyze the feasibility of the intersection of the two conditions:
| Condition | Solution Interval |
|---|---|
| $x < 5$ | $(-\infty, 5)$ |
| $x > 7$ | $(7, \infty)$ |
| Intersection ($P \cap Q$) | $\emptyset$ (Empty Set) |
There exists no real number $x$, and consequently no natural number $x$, that is strictly less than 5 and strictly greater than 7 at the same time [By the Law of Contradiction in Classical Logic].
Step 4: Visualization of the Number Line
Conclusion
Since the intersection of the sets representing the conditions is null, the set $A$ contains no elements. By definition, a set that contains no elements is termed a null set or empty set, denoted by $\emptyset$ or $\{\}$.
Final Solution: The set $\{x : x \in \mathbb{N}, x < 5 \text{ and } x > 7\}$ is an example of a null 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(ii): Which of the following are examples of the null set (ii) Set of even prime numbers
- 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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