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Section 1.4 Operations with sets – Union, Intersection and Complement
A universal set for a particular problem is a set which contains all the elements of all the sets in the problem.
A universal set is often denoted by a capital U, but sometimes the Greek letter ξ (xee) is
used.
In this section we will create subsets of a given universal set and use set operations to create new subsets of the universal set.
There are three set operations we will learn in this section.
Complement: The complement of a set A is symbolized by A’ and it is the set of all elements in the universal set that are not in A.
Intersection: The intersection of sets A and B is symbolized by 𝐴 ∩ 𝐵 and is the set containing all of the elements that are common to both set A and set B.
Union: The union of set A and B is symbolized 𝐴 ∪ 𝐵 and is the set containing all the elements that are elements of set A or of set B or that are in both Sets A and B.
Here is a quick example to illustrate the 3 definitions.
Example: Let U be a universal set and A and B be subsets of U defined as follows.
U = {1,2,3,4,5} A = {1,2,3} B = {2,3,4}
Find A’
A’ is all of the elements in the Universal set that are not in set A.
Answer: A’ = {4,5}
Find 𝑨 ∩ 𝑩 (This is asking me to find all of the elements that A and B have in common.)
Answer: 𝐴 ∩ 𝐵 = {2,3}
Find 𝑨 ∪ 𝑩 (This is asking me to list all of the elements in A followed by all of the elements in B, then delete any elements that are written twice.)
𝐴 ∪ 𝐵 = {1,2,3,2,3,4}
Answer: 𝐴 ∪ 𝐵 = {1,2,3,4}
Example: Let U be a universal set and A and B be subsets of U defined as follows.
U = {a,b,c,d,e,f} A = {a,b,c} B = {c,d,e}
Find 𝑨′ ∩ 𝑩
First I need to find A’, which is all of the elements in U that aren’t in set A.
A’ = {d,e,f}
Now I can intersect the two sets.
𝐴′^ ∩ 𝐵 = {d,e,f} ∩ {c,d,e} (now find what the two sets have in common)
Answer: {d,e}
Find 𝑨 ∪ 𝑩′
First I need to find B’
B’ = {a,b,f}
A ∪ B′ = {a,b,c} ∪ {a,b,f} (put all 6 elements in a big set then delete the duplicates)
= {a,b,c,a,b,f}
Answer: {a,b,c,f}
#1-10: Find the following sets.
U = {a,b,c,d,e} A = {c,d,e} B = {a,c,d}
A’ 2) B’
𝐴 ∪ 𝐵 4) 𝐴′ ∪ 𝐵′
𝐴 ∩ 𝐵 6) 𝐴′ ∩ 𝐵′
𝐴′ ∩ 𝐵 8) 𝐴 ∩ 𝐵′
𝐴′ ∪ 𝐵 10) 𝐴 ∪ 𝐵′
Find (𝑩 ∪ 𝑪)′
I have to work on the inside of the parenthesis first.
So I will first find: 𝐵 ∪ 𝐶
𝐵 ∪ 𝐶 = {c,d,e} ∪ {d,e,f}
𝐵 ∪ 𝐶 = {c,d,e,d,e,f}
𝐵 ∪ 𝐶 ={c,d,e,f}
Now I can do the complement.
I can replace the inside of the parenthesis with {c,d,e,f} and proceed to find its complement.
(B ∪ C)′ = (c,d,e,f)’ (my answer will be all the elements of set U that are not in this set.)
Answer: {a,b}
Find 𝑨 ∪ (𝑩 ∪ 𝑪)′
First I need to simplify the parenthesis (B ∪ C)′ I just figured out that (B ∪ C)′ = {a,b}, so I will use the work I have already done
A ∪ (B ∪ C)′
= A ∪ {a, b}
= {a,b,c} ∪ {a,b}
= {a,b,c,a,b}
Answer: {a,b,c}
Find 𝐴′ ∩ (𝐵 ∩ 𝐶′)
I need to simplify the inside of the parenthesis first.
(𝐵 ∩ 𝐶′)
= {c,d,e} ∩ {a,b,c}
= {c}
𝐴′ ∩ (𝐵 ∩ 𝐶′)
= A’ ∩ {𝑐}
= {d,e,f} ∩ {c}
Answer: ∅ (empty set)
#21-32: Find the following sets.
U = {1,2,3,4,5,6} A = {1,2,3} B = {2,3,4} C = {1,5}
𝐴 ∩ 𝐶 22) 𝐵 ∩ 𝐶
𝐴 ∪ 𝐶 24) 𝐵 ∪ 𝐶
𝐴 ∩ 𝐵 ∪ 𝐶 26) 𝐴 ∪ 𝐵 ∩ 𝐶
𝐵 ∪ 𝐶 ∩ 𝐴 28) 𝐵 ∩ 𝐴 ∪ 𝐶
𝐴′ ∩ 𝐵 30) 𝐴 ∩ 𝐵′
𝐴′ ∪ 𝐵 ∩ 𝐶′ 32) 𝐵′ ∩ 𝐴 ∪ 𝐶′
#33-44: Find the following sets.
U = {a,b,c,d} A = {a,b,c} B = {b,c,d} C = {a,d}
𝐴 ∩ 𝐶′ 34) 𝐵′ ∩ 𝐶
𝐴′ ∪ 𝐶′ 36) 𝐵′ ∪ 𝐶′
𝐴′ ∩ 𝐵 ∪ 𝐶′ 38) 𝐴′ ∪ 𝐵′ ∩ 𝐶
𝐵′ ∪ 𝐶′ ∩ 𝐴 40) 𝐵′ ∩ 𝐴′ ∪ 𝐶
𝐴′ ∩ 𝐵′ 42) 𝐴 ∩ 𝐵′
𝐴′ ∪ 𝐵′ ∩ 𝐶′ 44) 𝐵 ∩ 𝐴′ ∪ 𝐶′
