Lesson on Subsets

Subset Example 1Example 1: Given A = {1, 2, 4} and B = {1, 2, 3, 4, 5}, what is the relationship between these sets?

We say that A is a subset of B, since every element of A is also in B. This is denoted by:

a-subset-b.gif

Venn diagram for the relationship between these sets is shown to the right.

 

Answer: A is a subset of B.


Another way to define a subset is: A is a subset of B if every element of A is contained in BBoth definitions are demonstrated in the Venn diagram above.

Subset Example 2Example 2: Given X = {a, r, e} and Y = {r, e, a, d}, what is the relationship between these sets?

We say that X is a subset of Y, since every element of X is also in Y. This is denoted by:

x-subset-y.gif

A Venn diagram for the relationship between these sets is shown to the right.

 

Answer: X is a subset of Y.


Example 3: Given P = {1, 3, 4} and Q = {2, 3, 4, 5, 6}, what is the relationship between these sets?

We say that P is not a subset of Q since not every element of P is not contained in Q. For example, we can see that 1 is_not_an_element_of_transparent.gifQ. The statement "P is not a subset of Q" is denoted by:

p-not-subset-q.gif

Note that these sets do have some elements in common. The intersection of these sets is shown in the Venn diagram below.

subset_example3.png

Answer: P is not a subset of Q.


The notation for subsets is shown below.

Symbol Meaning
is_a_subset_of.gif is a subset of
is_not_a_subset_of.gif is not a subset of

Subset Example 4Example 4: Given A = {1, 2, 3, 4, 5} and B = {3, 1, 2, 5, 4}, what is the relationship between A and B?

Analysis: Recall that the order in which the elements appear in a set is not important. Looking at the elements of these sets, it is clear that:

a-subset-b_0.gif

b-subset-a.gif

a-equaivalent-to-b.gif

Answer: A and B are equivalent.


Definition: For any two sets, if is_a_subset_of-small.gif B  and  is_a_subset_of-small.gif A, then A = B. Thus A and B are equivalent.

Example 5: List all subsets of the set C = {1, 2, 3}.

Answer: 

 

Subset List all possible combinations of elements...
D = {1} one at a time
E = {2} one at a time
F = {3} one at a time
G = {1, 2} two at a time
M = {1, 3} two at a time
N = {2, 3} two at a time
P = {1, 2, 3} three at a time
Ø The null set has no elements.

Looking at example 5, you may be wondering why the null set is listed as a subset of C. There are no elements in a null set, so there can be no elements in the null set that aren't contained in the complete set. Therefore, the null set is a subset of every set. You may also be wondering: Is a set a subset of itself? The answer is yes: Any set contains itself as a subset. This is denoted by:

is_a_subset_of-small.gif A.

A subset that is smaller than the complete set is referred to as a proper subset. So the set {1, 2} is a proper subset of the set  {1, 2, 3} because the element 3 is not in the first set. In example 5, you can see that G is a proper subset of C, In fact, every subset listed in example 5 is a proper subset of C, except P. This is because and C are equivalent sets (P = C). Some mathematicians use the symbol is_a_subset_of-small_version2.gif to denote a subset and the symbol is_a_subset_of-small.gif to denote a proper subset, with the definition for proper subsets as follows:

If A is_a_subset_of-small_version2.gif B, and A  B, then A is said to be a proper subset of B and it is denoted by A is_a_subset_of-small.gif B.

While it is important to point out the information above, it can get a bit confusing, So let's think of subsets and proper subsets this way:

Subsets and Proper Subsets
The set {1, 2} is a proper subset of the set {1, 2, 3}.
The set {1, 2, 3} is a not a proper subset of the set  {1, 2, 3}.

Do you see a pattern in the examples below?

Example 6: List all subsets of the set R = {x, y, z}. How many are there?

Subsets
D = {x}
E = {y}
F = {z}
G = {x, y}
H = {x, z}
J = {y, z}
K = {x, y, z}
Ø

Answer: There are eight subsets of the set R = {x, y, z}.


Example 7: List all subsets of the set C = {1, 2, 3, 4}. How many are there?

Subsets
D = {1} M = {2, 4}
E = {2} N = {3, 4}
F = {3} O = {1, 2, 3}
G = {4} P = {1, 2, 4}
H = {1, 2} Q = {1, 3, 4}
J = {1, 3} R = {2, 3, 4}
K = {1, 4} S = {1, 2, 3, 4}
L = {2, 3} Ø

Answer: There are 16 subsets of the set C = {1, 2, 3, 4}.


In example 6, set R has three (3) elements and eight (8) subsets. In example 7, set C has four (4) elements and 16 subsets. To find the number of subsets of a set with n elements, raise 2 to the nth power: That is:

The number of subsets in set A is 2, where n is the number of elements in set A.


L e s s o n   S u m m a r y

Subset: A is a subset of B: if every element of A is contained in B. This is��denoted by is_a_subset_of-small.gif B.

Equivalent Sets: For any two sets, if is_a_subset_of-small.gif B  and  is_a_subset_of-small.gif A, then A = B.

Null set: The null set is a subset of every set.

Sets and subsets: Any set contains itself as a subset. This is denoted by is_a_subset_of-small.gif A.

Proper Subsets: If A is_a_subset_of-small_version2.gif B, and A  B, then A is said to be a proper subset of B and it is denoted by A is_a_subset_of-small.gif B.

Number of Subsets: The number of subsets in set A is 2, where n is the number of elements in set A.


Exercises

Directions: Read each question below. Select your answer by clicking on its button. Feedback to your answer is provided in the RESULTS BOX. If you make a mistake, rethink your answer, then choose a different button.

1. Which of the following is a subset of set G?

 G = {d, a, r, e}
 

 
 = {e, a, r}
 Y = {e, r, a}
 = {r, e, d}
 All of the above.

RESULTS BOX:
 

2. Which of the following statements is true?
 
 
 {vowels} is_a_subset_of-small.gif {consonants}
 {consonants} is_a_subset_of-small.gif {vowels}
 {vowels} is_a_subset_of-small.gif {alphabet}
 None of the above.

RESULTS BOX:
 

3. Which of the following is NOT a subset of set A?

A = {2, 3, 5, 7, 11}


 
 
 B = {3, 5, 2, 7}
 = {2, 3, 7, 9}
 = {7, 2, 3, 11}
 All of the above.

RESULTS BOX:
 

4. How many subsets will the set below have?

 T = {Monday, Tuesday, Wednesday, Thursday, Friday}
 

 
 5
 10
 32
 None of the above.

RESULTS BOX:
 

5. If R = {whole numbers < 5} and S = {4, 2, 0, 3, 1}, then which of the following statements is true?
 
 
 R = S
 has more elements than S.
 is null.
 None of the above.

RESULTS BOX:
 

IXL