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Q3(iv):
State whether each of the following set is finite or infinite: (iv) The set of animals living on the earth
Solution :
Step 1: Formal Definition of Finite and Infinite Sets
In set theory, a set $S$ is defined as finite if the cardinality of the set, denoted by $|S|$ or $n(S)$, is a non-negative integer. That is, the elements can be counted and the process of counting terminates [Per Zermelo-Fraenkel Set Theory Axioms]. Conversely, a set is infinite if it is not finite; it cannot be put into a one-to-one correspondence with the set $\{1, 2, 3, \dots, n\}$ for any natural number $n$.
Step 2: Analysis of the Given Set
The set under consideration is $A = \{x \mid x \text{ is an animal living on the earth}\}$. To determine the nature of $A$, we evaluate the constraints imposed by biological and physical reality:
- Biological Constraint: The number of living organisms at any specific discrete time $t$ is determined by the total biomass and population dynamics of all extant species.
- Temporal Constraint: While the population of animals fluctuates due to birth, death, and extinction rates, at any given moment in time, the number of individual animals inhabiting Earth is a finite, albeit large, integer.
- Mathematical Limit: Even if the exact count is difficult to calculate due to ecological complexity, the set does not satisfy the requirements of an infinite set (it does not contain an infinite sequence of distinct elements). It is bounded by the physical space and resources of the planet.
Step 3: Logical Deduction
Since the set of animals living on Earth at any time $t$ has a cardinal number $N$, where $N \in \mathbb{N}_0$, and there exists a definite upper bound to the number of biological entities supported by the biosphere, we conclude that the elements can be enumerated, theoretically, in finite time [By the Finite Set definition].
Step 4: Final Classification
Given that the population of animals on Earth is a bounded value, the set is finite.
Final Solution: The set of animals living on the earth is a finite 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(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(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
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