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

We speak that A is a subset of B, because every aspect of A is additionally in B. This is denoted by:


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

You are watching: List all the subsets of {3, 4, 7}.


Answer: A is a subset of B.

Another way to specify a subset is: A is a subset of B if every element of A is consisted of in B. Both definitions are prove in the Venn diagram above.

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

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


A Venn diagram because that the relationship in between these sets is displayed 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 in between these sets?

We to speak that P is not a subset of since not every aspect of P is not contained in Q. For example, we deserve to see that 1 

Q. The statement "P is not a subset the Q" is denoted by:


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


Answer: P is not a subset of Q.

The notation for subsets is shown below.

is a subset of
is no a subset of

Example 4: Given A = 1, 2, 3, 4, 5 and B = 3, 1, 2, 5, 4, what is the partnership between A and B?

Analysis: Recall that the order in i m sorry the elements appear in a collection is not important. Looking at the facets of these sets, the is clear that:




Answer: A and B are equivalent.

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

Example 5: List every subsets of the set C = 1, 2, 3.



SubsetList all feasible combinations that elements...
D = 1one in ~ a time
E = 2one at a time
F = 3one at a time
G = 1, 2two at a time
M = 1, 3two at a time
N = 2, 3two in ~ a time
P = 1, 2, 3three at a time
ØThe null set has no elements.

Looking at example 5, you might be wonder why the null set is provided as a subset of C. There space no elements in a null set, therefore there deserve to be no facets in the null set that aren"t had in the complete set. Therefore, the null set is a subset that every set. You may also be wondering: Is a set a subset the itself? The answer is yes: Any set contains itself together a subset. This is denoted by:

A  A.

A subset the is smaller sized than the complete set is described as a proper subset. So the collection 1, 2 is a suitable subset that the collection 1, 2, 3 since the element 3 is no in the very first set. In example 5, you can see that G is a proper subset of C, In fact, every subset detailed in instance 5 is a appropriate subset of C, except P. This is because P and C are equivalent sets (P = C). Part mathematicians usage the symbol  to represent a subset and also the symbol  to denote a suitable subset, through the definition for appropriate subsets as follows:

If A  B, and A ≠ B, then A is said to it is in a proper subset of B and the is denoted by A  B.

While the is crucial to point out the info above, that can gain a little bit confusing, therefore let"s think of subsets and proper subsets this way:

Subsets and Proper Subsets
The set 1, 2 is a appropriate subset the the set 1, 2, 3.
The collection 1, 2, 3 is a no a appropriate subset the the set 1, 2, 3.

Do you view a pattern in the instances below?

Example 6: List every subsets of the set R = x, y, z. How numerous are there?

D = x
E = y
F = z
G = x, y
H = x, z
J = y, z
K = x, y, z

Answer: There room eight subsets that the set R = x, y, z.

Example 7: List every subsets of the set C = 1, 2, 3, 4. How countless are there?

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

Answer: There space 16 subsets that the set C = 1, 2, 3, 4.

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

The variety of subsets in collection A is 2n , wherein 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 facet of A is consisted of in B. This is denoted by A  B.

Equivalent Sets: For any kind of two sets, if A  B and  B  A, then A = B.

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

Sets and also subsets: Any set contains itself together a subset. This is denoted by A  A.

Proper Subsets: If A  B, and A ≠ B, then A is claimed to be a suitable subset of B and the is denoted by A  B.

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


Directions: review each question below. Pick your price by clicking on its button. Feedback to your answer is noted in the outcomes BOX. If you make a mistake, rethink your answer, then select a various button.

1.Which the the complying with is a subset the set G?

 G = d, a, r, e 

  X = e, a, r Y = e, r, a Z = r, e, d All of the above.


2.Which that the adhering to statements is true? 
  vowels  consonants consonants  vowels vowels  alphabet None the the above.


3.Which that the following is no a subset the set A?

A = 2, 3, 5, 7, 11

  B = 3, 5, 2, 7 C = 2, 3, 7, 9 D = 7, 2, 3, 11 All of the above.


4.How plenty of subsets will certainly the set below have?

 T = Monday, Tuesday, Wednesday, Thursday, Friday 

  5 10 32 None the the above.



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If R = {whole number S = 4, 2, 0, 3, 1, climate which that the complying with statements is true? 
  R = S R has more elements than S. S is null. None that the above.