Subset, Superset, Universal Set and Complement of a Set

If set A is a subset of set B, then it is written as A ⊂ B. If it is not, then it's written as A ⊄ B.




Let's say you have 2 sets, A and B. If A ⊂ B, it means that ALL the elements of set A are also found in set B.

For example, if set A = {1, 2, 3} and set B = {1, 2, 3, 4}. A ⊂ B, since all the elements of C are found in D. Conversely, B ⊃ A, meaning that B is a superset of A. That means B contains all the elements of A, and then some more.

Let's say you have 2 other sets, C and D. If C is not a subset of D (also written as C ⊄ D), then one or more of set C's elements are not in D.

For example, if set C = {1, 3, 5} and D = {3, 5, 6}, then C ⊄ D.

How many subsets can you have for a given set?


where n(A) is the number of elements of set A. For example, if a given set Z = {0, 1}, then,



Note that an empty is also a subset of pretty much every set, including itself.


Representing set A ⊂ B as a Venn Diagram

If A ⊂ B, then set A can be drawn as being inside set B.

E.g.


Example 1
Given that A = {2, 3, 4, 5} and B = {x : x is an integer and 1 ≤ x ≤ 8}.
(a) By using the symbol ⊂, write a relation between A and B.
(b) Draw a Venn diagram to represent the relationship between the two sets.

Solution
(a) A ⊂ B
(b)


Example 2
The Venn diagram shows four sets P, Q, R and S. Write a relation between


(a) P and Q,
(b) Q and S,
(c) S and R,
(d) Q and R,
(e) P and P

Solution
(a) P ⊂ Q
(b) Q ⊂ S
(c) R ⊂ S
(d) Q ⊂ R
(e) P ⊂ P (since all elements of P are in P)


Universal Set

A universal set sounds like just what it is, a set containing all elements contained in your problem. It's basically the superset to all the sets in your problem.

The universal set is usually represented by the symbol U.

Example
Given that the universal set U = {x : 1 ≤ x ≤ 9, x is an integer},

A = {numbers that are a multiple of 3}, and
B = {factors of 8}.

(a) List all the elements of set A and set B
(b) Draw a Venn diagram to represent the relationship between U, A, and B.

Solution
(a) A = {3, 6, 9}, B = {1, 2, 4, 8}
(b)



Complement Set

A complement of set A is the set that contains all the elements that are not in set A. In a Venn diagram, it would look something like this.



The universal set contains all elements in both the shaded and unshaded area. Set A is the unshaded circle, while its complement, set A' (read as A prime), is represented by the shaded area.

Example
Given that universal set U = {factors of 100}, A = {perfect squares}, and B = {two-digit numbers}.

(a) List the elements of A' and B'.
(b) Draw a Venn diagram to represent (i) A' and U, (ii) B' and U.

Solution
U = {1, 2, 4, 5, 10, 20, 25, 50, 100}

(a)
A = {1, 4, 25, 100}
A' = {2, 5, 10, 20, 50}
B = {10, 20, 25, 50}
B' = {1, 2, 4, 5, 100}

(bi)


(bii)




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Complement of a universal set... 
What would be the complement of a universal set? Damien: By definition, the complement of a given set A refers to the set of elements outside A. …

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how do you do them? Damien: Do you have a sample problem you're having problems with? Maybe show what you've done so far?

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lets say A = {1,{2,3}} we can say 1 is an element of A right? and also {2,3} an element of A. but is 2 it self is an element of A? Damien: Hi …

Mom Teacher Not rated yet
Why is an empty set a subset of a set? Damien: It's by definition, i.e. how we give meaning to it. If you think about it, any set (including an empty …

modern college algebra I. set concepts a. VENN DIAGRAM Not rated yet
we have to draw a venn diagram using this given sets. U ={ x, y , z , 1 , 2 , 3 } X ={ 1,2,3 } Y={x,y,3} A={z} i want to know where will i put the …

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If you have the universe described as U={x|x=5n and n is an element of N} and A={x|x=2n-1 where n is an element of N} find the complement of A. My solution: …

I still cant understand.... Not rated yet
I just cant understand what subset is. Damien: Let's say you have the complete "set" of the "Lord of the Rings" DVD collection (that is, DVDs 1, 2 and …

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