Question

If U = \left\\{ {1,2,3,4,5,6,7,8,9} \right\\},A = \left\\{ {1,2,3,4} \right\\},B = \left\\{ {2,4,6,8} \right\\},C = \left\\{ {3,4,5,6} \right\\}
Find
i.$A'$
ii.$B'$
iii.$(A \cup C)'$
iv.$(A \cup B)'$
v.$\left( {A'} \right)'$
vi.$\left( {B - C} \right)'$

Answer 1

To solve the given type of function such that we need to remind the fact that $A' = U - A$ and substituting the given set we get the required result and the fact that the union is the collection of all the elements of the sets but.

Complete step-by-step answer:
i.$A'$
Using the fact that the term
$A' = U - A$
U = \left\\{ {1,2,3,4,5,6,7,8,9} \right\\},A = \left\\{ {1,2,3,4} \right\\}
Substituting the value
A' = \left\\{ {1,2,3,4,5,6,7,8,9} \right\\} - \left\\{ {1,2,3,4} \right\\}
Hence, after subtraction we get
A' = \left\\{ {1,2,3,4,5,6,7,8,9} \right\\} - \left\\{ {1,2,3,4} \right\\} = \\{ 5,6,7,8,9\\}
Hence, the above represents the required result

ii.$B'$
Using the fact that the term
$B' = U - B$
U = \left\\{ {1,2,3,4,5,6,7,8,9} \right\\},B = \left\\{ {2,4,6,8} \right\\}
Substituting the value
B' = \left\\{ {1,2,3,4,5,6,7,8,9} \right\\} - \left\\{ {2,4,6,8} \right\\}
Hence, after subtraction we get
A' = \left\\{ {1,2,3,4,5,6,7,8,9} \right\\} - \left\\{ {2,4,6,8} \right\\} = \left\\{ {1,3,5,7,9} \right\\}
Hence, the above represents the required result

iii.The third part can also be determined as
$(A \cup C)'$
We have given that
A = \left\\{ {1,2,3,4} \right\\},C = \left\\{ {3,4,5,6} \right\\}
A \cup C = \left\\{ {1,2,3,4} \right\\} \cup \left\\{ {3,4,5,6} \right\\}
Hence, we get
A \cup C = \left\\{ {1,2,3,4} \right\\} \cup \left\\{ {3,4,5,6} \right\\} = \\{ 1,2,3,4,5,6\\}
$(A \cup C)' = U - (A \cup C)$
Substituting the respective value, we get
$(A \cup C)' = \\{ 1,2,3,4,5,6,7,8,9\\} - \\{ 1,2,3,4,5,6\\} = \\{ 7,8,9\\}$
Hence, above represents the required result

iv.$(A \cup B)'$
We have given that
A = \left\\{ {1,2,3,4} \right\\},B = \left\\{ {2,4,6,8} \right\\}
A \cup B = \left\\{ {1,2,3,4} \right\\} \cup \left\\{ {2,4,6,8} \right\\}
Hence, we get
A \cup B = \left\\{ {1,2,3,4} \right\\} \cup \left\\{ {2,4,6,8} \right\\} = \\{ 1,2,3,4,6,8\\}
$(A \cup B)' = U - (A \cup B)$
Substituting the respective value, we get
$(A \cup B)' = \\{ 1,2,3,4,5,6,7,8,9\\} - \\{ 1,2,3,4,6,8\\} = \\{ 5,7,9\\}$
Hence, above represents the required result

v.$\left( {A'} \right)'$
Now we need to find $\left( {A'} \right)' = A$
A = \left\\{ {1,2,3,4} \right\\}
The above represent the required result.
Such that
$A' = U - A$
U = \left\\{ {1,2,3,4,5,6,7,8,9} \right\\}
A = \left\\{ {1,2,3,4} \right\\}
Substituting the value, we get
A' = \left\\{ {1,2,3,4,5,6,7,8,9} \right\\} - \\{ 1,2,3,4\\} = \\{ 5,6,7,8,9\\}
Now, $\left( {A'} \right)'$
$\left( {A'} \right)' = U - A'$
$\left( {A'} \right)' = \\{ 1,2,3,4,5,6,7,8,9\\} - \\{ 5,6,7,8,9\\} = \\{ 1,2,3,4\\}$
Hence the above represent the required result.

vi.$\left( {B - C} \right)'$
To solve such type of question we need to find the value which can be determined as
$\left( {B - C} \right)' = U - \left( {B - C} \right)$
That means we need to find the value of $(B - C)$
That can be simplified as
$(B - C) = B - (B \cap C)$
Intersection is the collection of the common element of the two sets
B = \left\\{ {2,4,6,8} \right\\},C = \left\\{ {3,4,5,6} \right\\}
Substituting the value, we get
B \cap C = \left\\{ {2,4,6,8} \right\\} \cap \left\\{ {3,4,5,6} \right\\} = \\{ 4,6\\}
B - (B \cap C) = \left\\{ {2,4,6,8} \right\\} - \\{ 4,6\\} = \\{ 2,8\\}
So, we get
$\left( {B - C} \right)' = U - \left( {B - C} \right)$
Substituting the value, we get
\left( {B - C} \right)' = \left\\{ {1,2,3,4,5,6,7,8,9} \right\\} - \left\\{ {2,8} \right\\} = \\{ 1,3,4,5,6,7,9\\}
The above equation represents the required result.

Note: To solve the given type of question $(A \cup B)' = A' \cap B'$ is the other method to determine the value of such type of question $(A \cup B)'$ , where union is the collection of all the elements and intersection denotes the value the elements of common elements between the sets.