Set Theory: Subsets
If every element of a set B is also a member of a set A, then we say B is a subset ofA. We use the symbol ⊂ to mean “is a subset of” and the symbol ⊄ to mean “is not a subset of”.
Example:
A = {1, 3, 5}, B = {1, 2, 3, 4, 5}
So, A ⊂ B because every element in A is also in B.
X = {1, 3, 5}, Y = {2, 3, 4, 5, 6}.
X ⊄ Y because 1 is in X but not in Y.
Note:
• Every set is a subset of itself i.e. for any set A, A ⊂ A
• The empty set is a subset of any set A i.e. Ø ⊂ A
• For any two sets A and B, if A ⊂ B and B ⊂ A then A = B
Example:
List all the subsets of the set Q = {x, y, z}
Solution:
The subsets of Q are { }, {x}, {y}, {z}, {x, y}, {x, z}, {y, z}and {x, y, z}
The number of subsets for a finite set A is given by the formula:
Number of subsets = 2 n(A)
where n(A) = number of elements in the finite set A
Example:
Q = {x, y, z}. How many subsets will Q have?
Solution:
n(Q) = 3
Number of subsets = 23 = 8
Example:
Draw a Venn diagram to represent the relationship between the sets. A = {1, 3, 5} and B = {1, 2, 3, 4, 5}
Solution:
Since A is a subset of B:
Step 1: Draw circle A within the circle B
Step 2 : Write down the elements in circle A.
Step 3 : Write down the remaining elements in circle B.