Recognition of Important Sets, Subsets & Power Sets – Class 8 Math with Examples and Quiz | Math With Raabi

 

Recognition of Important Sets, Subsets and Power Sets ✨

In mathematics, sets are essential tools used to represent groups of numbers or objects. Some sets are so commonly used that they have special names and symbols. Understanding these notations and relationships helps us in solving more advanced mathematical problems. 📘

Before learning about subsets and power sets, make sure you have read the previous topic Understanding Sets in Mathematics for a complete introduction to the concept of sets. 🔗

Important Sets and Their Notations 🧮

Set of Natural Numbers: The set containing all positive counting numbers. Denoted by N.

N = {1, 2, 3, 4, 5, ...}

Set of Whole Numbers: The set consisting of zero and all natural numbers. Denoted by W.

W = {0, 1, 2, 3, 4, ...}

Set of Integers: Includes all whole numbers and their negatives. Denoted by Z.

Z = {..., -3, -2, -1, 0, 1, 2, 3, ...}

Set of Rational Numbers: Numbers that can be expressed as fractions p/q, where p and q are integers and q ≠ 0. Denoted by Q.

Q = { x | x = p/q, p, q ∈ Z and q ≠ 0 }

Set of Even Numbers: Numbers divisible by 2. Denoted by E.

E = {0, ±2, ±4, ±6, ...}

Set of Odd Numbers: Numbers not divisible by 2. Denoted by O.

O = {±1, ±3, ±5, ±7, ...}

Set of Prime Numbers: Numbers greater than 1 that are divisible only by 1 and themselves. Denoted by P.

P = {2, 3, 5, 7, 11, 13, ...}

Subsets 🌟

If every element of one set A is also an element of another set B, then A is said to be a subset of B. This is represented as A ⊆ B.

Example: If A = {1, 2, 3, 4} and B = {1, 2, 3, 4, 5, 6}, then A is a subset of B because every element of A is present in B. ✅

If a set has n elements, the total number of subsets of that set is 2ⁿ.

Example 1: Write all the subsets of {1, 2}.
Solution: 2² = 4 subsets → ∅, {1}, {2}, {1, 2}.

Example 2: Write all the subsets of {1, 3, 5}.
Solution: 2³ = 8 subsets → ∅, {1}, {3}, {5}, {1,3}, {1,5}, {3,5}, {1,3,5}.

Example 3: Write all the subsets of {a, b, c, d}.
Solution: 2⁴ = 16 subsets → ∅, {a}, {b}, {c}, {d}, {a,b}, {a,c}, {a,d}, {b,c}, {b,d}, {c,d}, {a,b,c}, {a,b,d}, {a,c,d}, {b,c,d}, {a,b,c,d}.

Proper and Improper Subsets 🧩

Proper Subset: A subset A of B is called a proper subset if A is not equal to B — that means at least one element of B is not in A. Denoted by A ⊂ B.

Improper Subset: A subset that is exactly equal to the set itself. Every set is an improper subset of itself because it contains all its elements.

Note:

• All subsets of a set except the set itself are proper subsets.
• Every set is an improper subset of itself.
• The empty set has no proper subsets.
• A singleton set has only one proper subset: the empty set.

Power Set ⚡

The Power Set of a set A is the set of all subsets of A and is denoted by P(A). If a set A has n elements, then P(A) will have 2ⁿ elements.

Example 4: Write the power set of A = {1, 2}.
Solution: P(A) = { ∅, {1}, {2}, {1, 2} }.

Example 5: Write the power set of B = {x, y, z}.
Solution: P(B) = { ∅, {x}, {y}, {z}, {x,y}, {x,z}, {y,z}, {x,y,z} }.

Remember: Number of elements in P(A) = 2ⁿ, where n is the number of elements in A.
Also note that {a} ∈ P(A) but a ∉ P(A). 💡

Practice Time 📝

1. Write two proper subsets of each of the following sets:
   • {0, 2}
   • {1, 4, 7}
   • {-1, 0, 1}
   • {2, 4, 6, 8}
   • {a, b, c}
2. Write the power set of the following sets:
   • {1, 2}
   • {1, 3, 5}
   • {a, b, c}
3. For each set below, state whether it is a subset of the given set:
   • A = {1, 2} ; B = {0, 1, 2, 3, 4}
   • A = {1, 3, 5} ; B = {1, 3, 5, 7, 9}
4. State whether each statement is True or False:
   • Every set has a power set.
   • The empty set has no proper subsets.

🎯 Recognition of Important Sets, Subsets & Power Sets — Quick Quiz

Test your knowledge of important sets, subsets, and power sets. 🌸

Q1: What is the set of all positive counting numbers called, and what symbol represents it?

✅ It is called the set of natural numbers and is represented by N.

Q2: If A = {1, 2, 3} and B = {1, 2, 3, 4, 5}, what type of set is A with respect to B?

✅ A is a subset of B since all elements of A are in B.

Q3: How many subsets can a set with 4 elements have?

✅ It has 2⁴ = 16 subsets.

Q4: What is the difference between a proper subset and an improper subset?

✅ A proper subset is not equal to the original set, while an improper subset is equal to the set itself.

Q5: What is the power set of A = {x, y}?

✅ P(A) = { ∅, {x}, {y}, {x, y} }

Try It Yourself:
If a set has 4 elements, how many elements are there in its power set?
Answer: 2⁴ = 16 elements. 🌟

🔙 Read Previous Topic: Understanding Sets

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