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Q2(iii):
Which of the following sets are finite or infinite (iii) \{1, 2, 3, . . .99, 100\}
Solution :
Step 1: Formal Definition of Finite and Infinite Sets
To determine the classification of the set $A = \{1, 2, 3, \dots, 99, 100\}$, we must apply the foundational definitions of Set Theory:
- Finite Set: A set is finite if it is either empty or consists of a definite, countable number of elements such that the process of counting the elements comes to an end [Per Cantor’s Set Theory].
- Infinite Set: A set is infinite if it is not finite; that is, the process of counting its elements never terminates.
Step 2: Analysis of the Cardinality of Set A
The set is defined as $A = \{x \in \mathbb{N} : 1 \leq x \leq 100\}$.
We represent the set in roster form and identify its boundaries:
| Lower Bound ($min$) | Upper Bound ($max$) | Common Difference ($d$) |
|---|---|---|
| 1 | 100 | 1 |
The cardinality of a finite arithmetic sequence is calculated using the formula: $n = \frac{a_n - a_1}{d} + 1$.
Substituting the given values:
$n = \frac{100 - 1}{1} + 1$
$n = 99 + 1$
$n = 100$
Step 3: Logical Deduction
Since the cardinality $|A| = 100$, and $100$ is a fixed, non-negative integer, the set $A$ contains a definite number of elements. The counting process for this set is guaranteed to terminate at the 100th element [By the Principle of Finiteness].
Visual Representation
Final Solution: The set {1, 2, 3, . . .99, 100} is a Finite Set because it possesses a finite cardinality of 100.
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 $x > 7$ \}
- 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(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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