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Q2(iv):
Which of the following sets are finite or infinite (iv) The set of positive integers greater than 100
Solution :
Step 1: Formal Definition of the Set
Let $A$ be the set defined by the condition that its elements are positive integers ($x \in \mathbb{Z}^+$) such that $x > 100$. Mathematically, this set is expressed in set-builder notation as:
$A = \{x : x \in \mathbb{Z}^+, x > 100\}$
[By the definition of positive integers, $\mathbb{Z}^+ = \{1, 2, 3, \dots\}$]
Step 2: Enumeration of Elements
By applying the roster method to the condition $x > 100$, we identify the members of the set sequentially:
$A = \{101, 102, 103, 104, 105, \dots\}$
[By the Well-Ordering Principle, we identify the least element as $101$, but there is no maximum element in the set of positive integers.]
Step 3: Analytical Determination of Cardinality
A set is defined as finite if its elements can be counted and terminated, or if it is an empty set. A set is infinite if the process of counting its elements never reaches an end [Per the definition of Infinite Sets in Zermelo-Fraenkel set theory].
| Property | Observation for Set $A$ |
|---|---|
| Lower Bound | $101$ |
| Upper Bound | Does not exist ($\infty$) |
| Cardinality ($|A|$) | $\aleph_0$ (Countably infinite) |
Visual Representation of the Infinite Progression
Step 4: Conclusion
Since the set $A$ continues indefinitely without a supremum, it is impossible to list or count all elements. Therefore, it satisfies the criteria for an infinite set.
Final Solution: The set of positive integers greater than 100 is an infinite 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(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(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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