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Union of sets

Module by: Sunil Kumar Singh. E-mail the author

We are familiar with basic algebraic operations. These basic mathematical operations, however, are not valid in all contexts. For example, algebraic operation such as addition has different details, when operated on vectors. Clearly, we expect that these operations will also be not same in the case of sets – which are collections and not individual elements.

Nevertheless, set operations bear resemblance to algebraic operation. For example, when we combine (not add) two sets, then the operation involved is called “union”. We can see that there is resemblance of the intent of addition, subtraction etc in the case of sets also.

Venn diagrams

Venn diagrams are pictorial representation of sets/subsets and relationship that the sets/subsets have among them. It helps us to analyze relationship and carry out valid set operations in a relatively easier manner vis – a – vis symbolic representation.

Universal set

Universal set is the largest set among collection of sets. Importantly, it is not the collection of everything as might be conjectured by the nomenclature. For example, "R", is universal set comprising of all real numbers. The rational numbers, integers and natural numbers are its subset. In other consideration, we can call integers as universal set. In that case, sets such as {1,2,3}, prime numbers, even numbers, odd numbers are subset of the universal set of integers.

The universal set is pictorially represented by a region enclosed within a rectangle on Venn diagram. For illustration, consider the universal set of English alphabets and universal set of first 10 natural numbers as shown in the top row of the figure

Figure 1: The universal set is represented by a region enclosed within a rectangle.
Universal set
 Universal set  (u5.gif)

Many times, however, we may not be required to list elements of a universal set. In such case, we represent the universal set simply by a rectangle and the symbol for universal set, “U”, in the corner. This is particularly helpful, where number of elements in universal set are very large.

The subsets of the universal set are represented by closed curves – usually circles. The subset of vowels (V) is shown here within the circle with the listing of elements. Note that we have not listed all the alphabets for universal set and used the symbol “U” in the corner only.

Figure 2: The subset of the universal set is represented by a closed curve – usually circle.
Subset
 Subset  (u6.gif)

Union of sets

Union works on two operands, each of which is a set. The operation is denoted by symbol " ". Now, the question is : what do we expect when two sets are combined? Clearly, we need to enlist all the elements of two sets in the resulting set.

Definition 1: Union of two sets
The union of sets “A” and “B” is a third set, which consists all the elements of two sets.

In symbol,

A B = { x : x A o r x B } A B = { x : x A o r x B }

The word “or” in the set builder form defining union is important. It means that the element “x” belongs to either “A” or “B”. The element may belong to both sets (common to two sets), but not necessarily. We can, therefore, infer that union set consists of :

  1. elements exclusive to A
  2. elements exclusive to B
  3. elements common to A and B

As a set includes only distinct elements, the common elements are represented only once in the union set. Thus, union set consists of elements of both sets without repeating an element. Now, the set is represented on Venn diagram as shown here.

Figure 3: The representation of union on Venn diagram
Union of two sets
 Union of two sets  (u1.gif)

For illustration of working with union, let us consider two sets of positive integers as given here,

A = { 1,2,3,4,5,6 } A = { 1,2,3,4,5,6 }

B = { 4,5,6,7,8 } B = { 4,5,6,7,8 }

The union of two sets is :

A B = { 1,2,3,4,5,6,4,5,6,7,8 } A B = { 1,2,3,4,5,6,4,5,6,7,8 }

But repetition of elements in a set does not change it. Hence, we need not repeat elements in the resulting union.

A B = { 1,2,3,4,5,6,7,8 } A B = { 1,2,3,4,5,6,7,8 }

Here, universal set is natural numbers. The representation of union of joint sets is shown in the figure. We can observe that very construction of union on Venn diagram ensures that elements are not repeated.

Figure 4: The representation of union on Venn's diagram
Union of two sets
 Union of two sets  (u2.gif)

Interpretation of union set

Let us examine the defining set of union :

A B = { x : x A o r x B } A B = { x : x A o r x B }

We consider an arbitrary element, say “x”, of the union set. Then, we interpret the conditional meaning as :

I f x A B , t h e n x A o r x B . I f x A B , t h e n x A o r x B .

Can we emphasize this conditional meaning in reverse order :

I f x A o r x B , t h e n x A B . I f x A o r x B , t h e n x A B .

Yes, we can agree with the second conditional meaning as well. We, therefore, conclude that the statements work in both ways. We write two statements together as :

x A B x A o r x B x A B x A o r x B

We can reach yet another conclusion by observing representation of union set on Venn diagram. Now, if an arbitrary element “x” does not belong to union set, then it is clear that it does not belong to the region represented by the union set on the Venn’s diagram. Hence,

Figure 5: The representation of union on Venn's diagram
Union of two sets
 Union of two sets  (u1.gif)

I f x does not belong A B x A and x B . I f x does not belong A B x A and x B .

The important thing to note here is the word “and” in place of “or” used before. Think about it. Here two conditions follow simultaneously. If an element does not belong to an union set, then it will not belong to either of individual sets simultaneously. Now, the next thing to consider is whether this conditional statement will be true other way round as well?

I f x A a n d x B x A B . I f x A a n d x B x A B .

Yes, we can agree to this statement. We, therefore, conclude that the statements work in both ways. We write two statements together with the help of two ways arrow sign as :

x A B x A and x B . x A B x A and x B .

Union of disjoint sets

Consider students in class X and class XI. Let us denote the respective sets as "T" for tenth and "E" for eleventh class. Clearly, union i.e. combination of two sets should include elements from each of the sets. Hence,

T E = students in class X and XI T E = students in class X and XI

This is a straight forward union of two sets. The resulting set comprises of all elements present in both the sets. Since it is not possible that students studying in class X are also students of XI, we are sure that the numbers of elements in the union is sum of numbers of students in each class. As there is no commonality between two sets, it is a union of two “disjoint” sets. We conclude here that union of two disjoint sets has no common elements.

Union with subset

The set “B” consists of all elements of its subset “A”. In other words, the elements of a subset “A” also belongs to the set “B”. The operation of union is combining elements of two sets. The union with a subset, therefore, consists of elements from both “A” and “B”. However, all elements of “A” are also the elements of “B”. Therefore, we find that union set is same as the superset “B”. Symbolically,

I f A B , then B A = B . I f A B , then B A = B .

We can check this deduction with the help of an example. Let us consider two sets as :

A = { 4,5,6 } A = { 4,5,6 }

B = { 1,2,3,4,5,6 } B = { 1,2,3,4,5,6 }

Here, we see that A B. Now,

B A = { 1,2,3,4,5,6,4,5,6 } = { 1,2,3,4,5,6 } = B B A = { 1,2,3,4,5,6,4,5,6 } = { 1,2,3,4,5,6 } = B

Figure 6: The representation of union with subset on Venn diagram
Union with subset
 Union with subset  (u3a.gif)

Multiple unions

If A 1 , A 2 , A 3 , , A n A 1 , A 2 , A 3 , , A n is a finite family of sets, then their unions, one after another, is denoted as :

A 1 A 2 A 3 A n A 1 A 2 A 3 A n

Important results

In this section we shall discuss some of the important characteristics/ deductions for the union operation.

Idempotent law

The literal meaning of the word “idempotent” is “unchanged when multiplied by itself”. Following the clue, the union of a set with itself is the set itself. This is an equivalent statement conveying the meaning of “idempotent” in the context of union. Symbolically,

A A = A A A = A

The union set consists of distinct elements and common elements taken once. Between two sets here, all elements are common. The union set consists of all elements of either set.

Identity law

The algebraic operators like addition and multiplication have defined identities, which does not change the other operand of the operator. For example, if we add “0” to a number, it remains same. Hence, “0” is additive identity. Similarly, “1” is multiplicative identity.

In the case of union, we find that union of a set with empty set does not change the set. Hence, empty set is union identity.

A φ = A A φ = A

As there is no element in empty set, union has same elements as that in “A”.

Law of U

All sets are subsets of universal set for a given context. We have seen that union with subset results in the set itself. Clearly, union of universal set with its subset will result in the universal set itself.

U A = U U A = U

Commutative law

In order to assess whether commutative property holds or not, we consider the example, used earlier. Let the sets be :

A = { 1,2,3,4,5,6 } A = { 1,2,3,4,5,6 }

B = { 4,5,6,7,8 } B = { 4,5,6,7,8 }

Then,

A B = { 1,2,3,4,5,6,4,5,6,7,8 } = { 1,2,3,4,5,6,7,8 } A B = { 1,2,3,4,5,6,4,5,6,7,8 } = { 1,2,3,4,5,6,7,8 }

B A = { 4,5,6,7,8,1,2,3,4,5,6 } = { 1,2,3,4,5,6,7,8 } B A = { 4,5,6,7,8,1,2,3,4,5,6 } = { 1,2,3,4,5,6,7,8 }

Thus, we see that order of operands with respect to the union operator is not differentiating. We can also appreciate this law on Venn diagram, which does not change by changing positions of sets across union operator.

Associative law

The associative property also holds with respect to union operator. We know that associative property is about changing the place of parentheses as here :

A B C = A B C A B C = A B C

The parentheses simply change the precedence of operation. On Venn diagram, union involving three sets appears same, irrespective of whether we apply union operation in a particular sequence.

Figure 7: Associative law
Union of three sets
 Union of three sets  (u4.gif)

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