# The Union of Two Sets — a Set Operation

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 AB. So if xis an element of A or of B or of both, then 1: is an element of AB.

⛲ Example 1. For the sets C={bear, camel, horse, dog, cat} and D={lion, elephant, horse, dog}, we would get CD={bear, camel, horse, dog, cat, lion, elephant}.
To see this using a Venn diagram, we would give each set a color. Then CD would be anything in the diagram with any color.

Union

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

AB={x|xA or xB}

[Important note] when we say “or” we always mean “inclusive or”. So if a∈A and a∈B, then a∈AB.

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

EF={-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}
GH={a,c,d,e,i,o,u}
②. I={1,3,5}, J={2,4,6}
IJ={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}
KL={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 ST. Using curly brace notion

ST={x: (xS) or (xT)}

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:

ST={x: (xS)∨(xT)}

⛲ Ex4: Unions of sets.
Suppose M={1,2,3}, N={1,3,5}, and O={2,3,4,5}.
Then:

MN={1,2,3,5},
MO={1,2,3,4,5}, and
NO={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

𝕌={1,2,3,4,5.6,7,8.9,10}
P={1,2,4,6}
Q={1,3,6,7,9}
R={}

determine each of the following. ①. PQ, ②. PR, ③. P̄∪Q, ④. (PQ).
Solution:
①. PQ={1,2,4,6}∪{1,3,6,7,9}={1,2,3,4,6,7,9}
②. PR={1,2,4,6}∪{}={1,2,4,6}.Note that PR=A.
③. To determine P̄∪Q, we must determine P̄.
P̄={3,5,7,8,9,10}
P̄∪Q={3,5,7,8,9,10}∪{1,3,6,7,9}
={1,3,5,6,7,8,9,10}

④. Determine (PQ) by first determining PQ and then find the complement of PQ.
PQ={1,2,3,4,6,7,9} from part ①
(PQ)={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 AB is read as “A union B”.

Symbolically: AB={x|xA or xB}

⛲ 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
Solution:
①. X={1,3,5} Y={1,2,3} XY={1,2,3,5}
②. E={a,e,i,o,u}, F={a,b,c}
EF={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}
GH={1,2,4,5,3,6,9,12,…}
GH={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} IJ={2,3,4,5,6,7,8,9}
IJ={x:x∈ℕ and 1<x<10} ⑤. K={1,2,3}, L
KL={1,2,3}

⛲ Example 7.
①. If C={5,7,8}, D={2,7,9,10,11} then CD={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 CD={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 ABC 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 VWZ.

VW={1,2,3,4,6,8}, (VW)∪Z={1,2,3,4,5,6,8}.
WZ={2,3,4,5,6,8}, V∪(WZ)={1,2,3,4,5,6,8}.
VZ={1,2,3,4,5,6}, W∪(VZ)={1,2,3,4,5,6,8}.

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