Readings for Session 4   

Subsets of a Set 

What is the relationship between the U.S. Senate and its Judiciary Committee?

 

In 2009, the two senators from Minnesota, Amy Klobucher and Al Franken were both members of the U.S. Senate’s Judiciary Committee, but Minnesota’s 7^th Congressional District representative, Collin Peterson, could not be a member of that committee.

 

        In order to be a member of the Judiciary Committee a person must first be a member of the U.S. Senate. This is an example of a collection of objects chosen from another collection. This leads to the concept of a subset.  

Subset: A set A is a subset of a set B if every element of A is also an element of B.

Notation: A ⊆ B  is read, “Set A is a subset of set B.”

 

Example:  In the paragraph above, the set of members of the U.S. Senate’s Judiciary Committee is a subset of the set of members of the U.S. Senate.

 

Example:  For A = {red, blue} and B = {red, white, blue},  A ⊆ B since every element of A is also an element of B.

 

Example: Let C = {a, b, c} and D = {b, c, d, e}. Then C ⊈ D, read “C is not a subset of D” since a is an element of C but not an element of  D. Also, D ⊈ C since e ∈ D and e ∉ C.

 

Example:  A ⊆ A and  ∅ ⊆ A. 
A set is a subset of itself since a set contains all its elements. Also, the empty set is a subset of every set, because every element in the empty set belongs to any set since the empty set has no elements.

 Listing Subsets:   List all the subsets of {a, b, c}.                 

Example:  The set {a, b, c} has 8 subsets.  They are:  
                   ∅, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, and {a, b, c}. 

Proper Subset:   A proper subset is a special type of subset.  There are two requirements for set A to be a proper subset of set B.  They are:

               1.  A is a subset of B, i.e., A ⊆ B   and
               2.  A is not equal to B, i.e., A ≠ B.

Notation: A ⊂ B is read, “Set A is a proper subset of set B.”

Example:  The set of members of the U.S. Senate’s Judiciary Committee is a subset of the set of members of the U.S. Senate since not every member of the U.S. Senate is on the Judiciary Committee.

Example:  For A = {red, blue} and B = {red, white, blue}, A ⊂ B since both requirements are met:
                         1.   A ⊆ B since red and blue are in both sets A and B; and
                         2.   A ≠ B since set B contains the element “white” but set A does not.

 Listing Proper Subsets:   List all the proper subsets of {a, b, c}.                 
 

Example:  The set {a, b, c} has 7 proper subsets.  They are:  
                      ∅, {a}, {b}, {c}, {a, b}, {a, c}, and {b, c}.

 

Note that {a, b, c} is not a proper subset of {a, b, c}. Also, note that there is always one less proper subset than there are subsets of a set since a set cannot be a proper subset of itself.

 

     

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