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

Hint: 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.