Set Theory: Subsets



In this lesson, we will learn
  • about subsets and proper subsets
  • the formula for the number of subsets
Related Topics: More Lessons on Sets

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. 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 = 2=  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.


Subset and Proper Subset
This video defines and give the notation used for subsets and proper subsets.



Learn about subset and proper subset.




Number of Subsets



Subsets of the Real Number System
Brief description of the set of real numbers and its subsets: naturals, wholes, integers, rationals, and irrationals.








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