krit.club logo

Sets - Finite and Infinite Sets-advanced

Grade 9CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

•

A set which is empty or consists of a definite number of elements is called a finite set. For example, the set of vowels in the English alphabet V={a,e,i,o,u}V = \{a, e, i, o, u\} is finite because it has exactly 55 elements.

•

The number of distinct elements in a finite set AA is called its cardinal number or order, denoted by n(A)n(A). If A={1,2,3}A = \{1, 2, 3\}, then n(A)=3n(A) = 3.

•

A set which is not finite is called an infinite set. The elements of an infinite set cannot be counted or listed completely, such as the set of natural numbers N={1,2,3,… }\mathbb{N} = \{1, 2, 3, \dots\}.

•

The empty set ∅\emptyset is considered a finite set because the number of elements in it is a definite whole number, specifically n(∅)=0n(\emptyset) = 0.

•

All infinite sets cannot be described in roster form. For example, the set of real numbers R\mathbb{R} cannot be described by a pattern with dots because its elements do not follow a simple discrete sequence.

•

Two finite sets AA and BB are called equivalent sets if they have the same cardinal number, i.e., n(A)=n(B)n(A) = n(B). All equal sets are equivalent, but all equivalent sets are not necessarily equal.

📐Formulae

n(A)=k, where k∈{0,1,2,3,… } (for finite sets)n(A) = k, \text{ where } k \in \{0, 1, 2, 3, \dots\} \text{ (for finite sets)}

A={x:x∈N}  ⟹  n(A)→∞ (infinite set)A = \{x : x \in \mathbb{N}\} \implies n(A) \to \infty \text{ (infinite set)}

If A⊂B and A is infinite, then B is also infinite.\text{If } A \subset B \text{ and } A \text{ is infinite, then } B \text{ is also infinite.}

If B is finite and A⊆B, then A must be finite.\text{If } B \text{ is finite and } A \subseteq B, \text{ then } A \text{ must be finite.}

💡Examples

Problem 1:

Determine if the set A={x:x∈Z,x2<25}A = \{x : x \in \mathbb{Z}, x^2 < 25\} is finite or infinite. If finite, find n(A)n(A).

Solution:

The condition given is x2<25x^2 < 25 where xx is an integer (Z\mathbb{Z}). Solving the inequality: −5<x<5-5 < x < 5 The integers satisfying this are x∈{−4,−3,−2,−1,0,1,2,3,4}x \in \{-4, -3, -2, -1, 0, 1, 2, 3, 4\}. Since we can list and count the elements, the set is finite. n(A)=9n(A) = 9.

Explanation:

To check if a set in set-builder form is finite, we solve the constraints to see if the solution set has a fixed number of elements.

Problem 2:

State whether the set of all concentric circles in a plane is finite or infinite.

Solution:

Let the center of the circles be (h,k)(h, k). The equation of a circle is (x−h)2+(y−k)2=r2(x-h)^2 + (y-k)^2 = r^2. Since the radius rr can be any positive real number (r>0r > 0), and there are infinitely many real numbers between any two values, there are infinitely many possible circles.

Explanation:

Geometric sets involving continuous parameters (like radius or length) are typically infinite because the parameter can take an infinite number of values.

Problem 3:

Consider the set B={x:x∈N and x is a prime number less than 100}B = \{x : x \in \mathbb{N} \text{ and } x \text{ is a prime number less than } 100\}. Is this set finite?

Solution:

The prime numbers less than 100100 are: {2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97}\{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97\}. Counting them, we find n(B)=25n(B) = 25. Since 2525 is a definite natural number, the set is finite.

Explanation:

Even if a set has many elements, as long as the count is a fixed whole number, it remains a finite set.