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Definition
Subsets
If the set B is made by taking some elements of set A, then set B is called a subset of set A.
Symbolically: B ⊂ A and A ⊃ B
Every element of B must also be an element of A for B to be a subset of A. Subsets include both proper and improper subsets.
Definition
Proper Subsets
If set B is a subset of set A and the number of elements of B is less than that of A, then set B is called a proper subset.
Hence, the empty set { } is a proper subset of all sets (except itself).
Example — Proper Subsets of M = {s, k, y}
All possible subsets of M = {s, k, y}:
No elements:
{ }
One element:
{s}
{k}
{y}
Two elements:
{s, k}
{s, y}
{k, y}
All 7 sets above are proper subsets of M, as none of them equal M itself.
Question Solving Method — Proper Subset
Given:
U = {2,4,6,8,10,12,14,16,18,20}
X = {2,4,6,8,10,12,14,16}
Y = {2,4,6,8}
Represent the relation between X and Y in a Venn diagram.
Step 1
Check if all elements of Y exist in X.
Y = {2, 4, 6, 8} → All elements are present in X = {2, 4, 6, 8, 10, 12, 14, 16}. ✓
Step 2
Compare the number of elements.
n(Y) = 4 < n(X) = 8 → Y has fewer elements than X. ✓
Step 3
Draw the Venn Diagram — Y circle is drawn inside X circle (Y ⊂ X):
U
X
Y
10
12
14
16
18
20
2
4
6
8
Step 4
Since every element of Y is in X, but Y ≠ X, Y is a proper subset of X.
Step 5
Write the conclusion using set notation.
∴ Y is the proper subset of X. Symbolically: Y ⊂ X
📋 Question 1 of 3
Definition
Improper Subsets
If set B is made by taking all elements of set A, then set B is called an improper subset of set A.
Symbolically: B ⊆ A and A ⊇ B
Equal sets are the improper subsets of one another.
Example — Improper Subset of M = {s, k, y}
The only improper subset of M = {s, k, y} is the set containing all its elements:
H = {s, k, y}
This subset H = {s, k, y} is an improper subset because it contains every element of M — making H equal to M.
Question Solving Method — Improper Subset
Given:
U = {1,2,3,4,5,6,7,8}
P = {2,4,6,8}
Q = {2,4,6,8}
Represent the relation between P and Q in a Venn diagram.
Step 1
Check if all elements of Q exist in P.
Q = {2, 4, 6, 8} → All elements are present in P = {2, 4, 6, 8}. ✓
Step 2
Compare the number of elements.
n(Q) = 4 = n(P) = 4 → Both sets have equal elements. ✓
Step 3
Draw the Venn Diagram — P and Q coincide (equal sets):
Step 4
Since P = Q, they are equal sets and hence improper subsets of one another.
Step 5
Write the conclusion using set notation.
∴ Q is the improper subset of P. Symbolically: Q ⊆ P and P ⊆ Q
📋 Question 1 of 3