Mastering Sets: ICSE Class 8th Maths (Selina Chapter 6)

Introduction

Hello Students! Today, we are going to explore the fundamental concepts of Sets from your Selina Concise Mathematics Chapter 6. A set is essentially a collection of well-defined objects or elements. In this blog, we will learn how to classify sets based on the number of elements they contain and how to compare two different sets. Understanding these basics is crucial for mastering higher-level algebra!

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Concept Explanation

In Mathematics, we classify sets into four primary types based on how many elements are inside them:

1. Finite Set: A set where the elements are countable. For example, the set of members in your family or natural numbers between 2 and 8.
2. Infinite Set: A set where the elements are uncountable or endless. For instance, the set of all living things on Earth or whole numbers greater than 1,000.
3. Singleton Set: A set that contains only one single element. An example would be “The President of India” or the whole number between 6 and 8.
4. Empty (or Null) Set: A set that contains no elements at all. It is denoted by the symbol ϕ (phi) or empty curly brackets { }.

When we compare two sets, we use these terms:

• Joint (Overlapping) Sets: Sets that have at least one common element.
• Disjoint Sets: Sets that have no elements in common.
• Equivalent Sets: Sets that have the same number of elements, even if the elements themselves are different.
• Equal Sets: Sets where every single element is exactly identical.

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Rules and Laws of Sets

To identify and work with sets correctly, remember these rules:

1. Countability Rule: If you can reach an end while counting elements, it is a Finite Set; if not, it is Infinite.
2. The Identity Rule: For two sets to be Equal, they must contain the exact same members in any order.
3. The Cardinality Rule: For sets to be Equivalent, only the count of elements must match (n(A)=n(B)).
4. The Subset Law: Every equal set is automatically an equivalent set, but equivalent sets are not always equal.

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Notes

• Important Distinction: A set containing zero {0} or a set containing the phi symbol {ϕ} is NOT an empty set. These are Singleton Sets because they contain one element (the number zero or the symbol phi).
• Empty Set Representation: Use either ϕ or { }, but never put the phi symbol inside the brackets like {ϕ} to represent a null set.

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Examples with Solutions

Example 1 — Find the elements of the set A={x:x∈Integer and x2−4x=0}.

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Solution — We need to solve the quadratic equation x2−4x=0:

1. Take x as a common factor: x(x−4)=0.
2. This means either x=0 or x−4=0.
3. Solving x−4=0 gives x=4.
4. The values of x are 0 and 4, which are both integers.

Ans. A={0,4}

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Example 2 — Represent the set B={x:x∈Integer and x2=9} in Roster Form.

Solution — We solve for x by taking the square root:

1. x2=9
2. x=±9
3. ​
4. x=3 or x=−3. Since both 3 and -3 are integers, both are included in the set.

Ans. B={3,−3}

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Example 3 — Determine if the set C={x:x∈N,x=n−3,n❤,n∈N} is an empty set.

Solution —

1. First, find values for n. Since n is a natural number (N) and n❤, n can be 1 or 2.
2. If n=1, then x=1−3=−2.
3. If n=2, then x=2−3=−1.
4. However, the rule states x must be a Natural Number. Since -2 and -1 are not natural numbers, there are no valid elements for this set.

Ans. Set C=ϕ (Empty Set)