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Q1(iv):
Which of the following are examples of the null set (iv) \{ $y$ : $y$ is a point common to any two parallel lines\}
Solution :
Step 1: Formal Definition of the Set
The set is defined by the set-builder notation $S = \{ y : y \text{ is a point common to any two parallel lines} \}$. To determine if this is a null set (an empty set, denoted as $\emptyset$ or $\{\}$), we must evaluate the existence of any element $y$ that satisfies the given condition.
Step 2: Geometric Analysis of Parallel Lines
By Euclidean geometry axioms, two lines are defined as parallel if and only if they lie in the same plane and do not intersect at any point, regardless of how far they are extended in either direction [Per Euclid's Fifth Postulate and the definition of Parallelism].
- Let $L_1$ and $L_2$ be two distinct lines in a plane.
- If $L_1 \parallel L_2$, then for every point $P$ such that $P \in L_1$, it is guaranteed that $P \notin L_2$.
- Conversely, for every point $Q$ such that $Q \in L_2$, it is guaranteed that $Q \notin L_1$.
Step 3: Visual Representation
Step 4: Logical Evaluation
Since the distance $d$ between two parallel lines is constant and $d > 0$, the set of intersection points must be empty. Mathematically:
$L_1 \cap L_2 = \emptyset$
Because there exists no point $y$ such that $y \in L_1$ and $y \in L_2$, the set $S$ contains no elements.
Conclusion
By the definition of a null set (a set containing zero elements), since the condition "point common to any two parallel lines" is impossible to satisfy, the set is indeed a null set.
Final Solution: The set $\{ y : y \text{ is a point common to any two parallel lines} \}$ is an example of a null set ($\emptyset$).
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(iii): Which of the following are examples of the null set (iii) \{ $x$ : $x$ is a natural numbers, $x < 5$ and $x > 7$ \}
- 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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