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 of A. 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. Ř ⊂  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.

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