**Set Operations**. We will need to be able to do some basic operations with sets.

The first operation we will consider is called the __union__ of sets. This is the set that we get when we combine the elements of two sets. The union of two sets, *A* and *B* is the set containing all elements of both *A* and B; the notation for *A* union *B* is *A*∪*B*. So if *x*is an element of *A* or of *B* or of both, then 1: is an element of *A*∪*B*.

⛲ Example 1. For the sets *C*={bear, camel, horse, dog, cat} and *D*={lion, elephant, horse, dog}, we would get *C*∪*D*={bear, camel, horse, dog, cat, lion, elephant}.

To see this using a Venn diagram, we would give each set a color. Then *C*∪*D* would be anything in the diagram with any color.

**Union**

Given two sets, *A* and *B*, we define their **union**, denoted *A*∪*B*, to be the set

*A*∪

*B*={

*x*|

*x*∈

*A*or

*x*∈

*B*}

[

__Important note__] when we say “or” we

**always**mean “inclusive or”. So if a∈

*A*and a∈

*B*, then a∈

*A*∪

*B*.

⛲ Ex2. If *E*={1,2,6,7,8} and *F*={-1,3,6,8},what is *E*∪*F*?

*E*∪

*F*={-1,1,2,3,6,7,8}.

⛲ Ex3. Find the union of each of the following pairs of sets.

①. *G*={a,e,i,o,u}, *H*={a,c,d}

②. *I*={1,3,5}, *J*={2,4,6}

③. *K*={*x*:*x* is a natural number and 1<*x*≤5} and *L*={*x*:*x* is a natural number and 5<*x*≤10}

🌟 Firstly, convert the given set in roster form, if it is not given in that. Then union of two sets, is the set which consists of all those elements which are either in *A* or in *B*.

Solution:

①. *G*={a,e,i,o,u},*H*={a,c,d}

⇒*G*∪*H*={a,c,d,e,i,o,u}

②. *I*={1,3,5}, *J*={2,4,6}

⇒*I*∪*J*={1,2,3,4,5,6}

③. *K*={*x*:*x* is a natural number and 1<*x*≤5}

⇒*K*=[2,3,4,5}

*L*={*x*:*x* is a natural number and 5<*x*≤10}

⇒ *L*={6,7,8,9,10}

*K*∪*L*={2,3,4,5}∪{6,7,8,9,10}={2,3,4,5,6,7,8,9,10}

[__Definition 1__] The union of two sets *S* and *T* is the collection of all objects that are in either set. It is written *S*∪*T*. Using curly brace notion

*S*∪

*T*={

*x*: (

*x*∈

*S*) or (

*x*∈

*T*)}

The symbol or is another Boolean operation, one that is true if either of the propositions it joins are true. Its symbolic equivalent is ∨ which lets us rewrite the definition of union as:

*S*∪

*T*={

*x*: (

*x*∈

*S*)∨(

*x*∈

*T*)}

⛲ Ex4: **Unions of sets**.

Suppose *M*={1,2,3}, *N*={1,3,5}, and *O*={2,3,4,5}.

Then:

*M*∪

*N*={1,2,3,5},

*M*∪

*O*={1,2,3,4,5}, and

*N*∪

*O*={1,2,3,4,5}

When performing set theoretic computations, you should declare the domain in which you are working. In set theory this is done by declaring a universal set.

⛲ Ex5: **Unions of Sets**

Given

*P*={1,2,4,6}

*Q*={1,3,6,7,9}

*R*={}

determine each of the following. ①.

*P*∪

*Q*, ②.

*P*∪

*R*, ③. P̄∪

*Q*, ④. (

*P*∪

*Q*).

Solution:

①.

*P*∪

*Q*={1,2,4,6}∪{1,3,6,7,9}={1,2,3,4,6,7,9}

②.

*P*∪

*R*={1,2,4,6}∪{}={1,2,4,6}.Note that

*P*∪

*R*=A.

③. To determine P̄∪

*Q*, we must determine P̄.

P̄∪

*Q*={3,5,7,8,9,10}∪{1,3,6,7,9}

={1,3,5,6,7,8,9,10}

④. Determine (

*P*∪

*Q*) by first determining

*P*∪

*Q*and then find the complement of

*P*∪

*Q*.

*P*∪

*Q*={1,2,3,4,6,7,9} from part ①

(

*P*∪

*Q*)={1,2,3,4,6,7,9}={5,8,10}

**Operations On Sets: Union Of Sets**

[__Definition 2__] The union of two sets *A* and *B* is the set whose elements are all of the elements in *A* or in *B* or in both.

The union of sets *A* and *B* denoted by *A*∪*B* is read as “*A* union *B*”.

Symbolically: *A*∪*B*={*x*|*x*∈*A* or *x*∈*B*}

⛲ Ex6. Find the union of each of the following pairs of sets:

①. *X*={1,3,5}; *Y*={1,2,3}

②. *E*={a,e,i,o,u}, *F*={a,b,c}

③. *G*={*x*:*x* is a natural number and multiple of 3}

*H*={*x*:*x* is a natural number less than 6}

④. *I*={*x*:*x* is a natural number and 1<*x*≤6}

*J*={*x*:*x* is a natural number and 6<*x*<10}
⑤. *K*={1,2,3}; *L*=Ø

Solution:

①. *X*={1,3,5} *Y*={1,2,3} *X*∪*Y*={1,2,3,5}

②. *E*={a,e,i,o,u}, *F*={a,b,c}

*E*∪*F*={a,b,c,e,i,o,u}

③. *G*={*x*:*x* is a natural number and multiple of 3}={3,6,9,…}

*H*={*x*:*x* is a natural number less than 6}={1,2,3,4,5,6}

*G*∪*H*={1,2,4,5,3,6,9,12,…}

*G*∪*H*={*x*:*x*=1,2,4,5 or a multiple of 3}

④. *I*={*x*:*x* is a natural number and 1<*x*≤6}={2,3,4,5,6}

*J*={*x*:*x* is a natural number and 6<*x*<10}={7,8,9}
*I*∪*J*={2,3,4,5,6,7,8,9}

*I*∪*J*={*x*:*x*∈ℕ and 1<*x*<10}
⑤. *K*={1,2,3}, *L*=Ø

*K*∪*L*={1,2,3}

⛲ Example 7.

①. If *C*={5,7,8}, *D*={2,7,9,10,11} then *C*∪*D*={2,5,7,8,9,10,11}

②. If *C*={*x*|*x*∈ℤ, and *x*≥3} and *D*={*x*|*x*∈ℤ, and *x*≥8}

then *C*∪*D*={*x*|*x*∈ℤ, *x*≥3}

Where ℤ denoted the set of integers.

**Union of 3 sets**

If *A* and *B* and *C* are sets, their union *A*∪*B*∪*C* is the set whose elements are those objects which appear in at least one of *A* or *B* or C.

⛲ Ex8. If *V*={l,2,3,4}, *W*={2,4,6,8} and *Z*={3,4,5,6},list the elements of the set *V*∪*W*∪*Z*.

*V*∪

*W*={1,2,3,4,6,8}, (

*V*∪

*W*)∪

*Z*={1,2,3,4,5,6,8}.

*W*∪

*Z*={2,3,4,5,6,8},

*V*∪(

*W*∪

*Z*)={1,2,3,4,5,6,8}.

*V*∪

*Z*={1,2,3,4,5,6},

*W*∪(

*V*∪

*Z*)={1,2,3,4,5,6,8}.