Question

Let S be the universal set, where: S = { 1 , 2 , 3 , ... , 18 , 19 , 20 } Let sets A , B , and C be subsets of S , where: Set A = { 1 , 6 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 18 , 19 , 20 } Set B = { 1 , 3 , 4 , 6 , 8 , 9 , 10 , 11 , 12 , 15 , 20 } Set C = { 3 , 6 , 7 , 8 , 9 , 10 , 11 , 13 , 14 , 17 } Find the number of elements in the set ( A ∪ B ∪ C ) n ( A ∪ B ∪ C ) = Find the number of elements in the set ( A ∩ B ∩ C ) n ( A ∩ B ∩ C ) =

Answer 1

Answer:

1. A B C=(6,10,11)

2.A B C =(1,3,4,5,7,8,9,12,13,14......20)

Answer 2

The answer is:$\to \bold{(A \cup B\cup C) \cap(A\cup B\cup C) = \bold{\{1,3,4,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20 \}} }\\\\\to \bold{(A \cap B\cap C) \cap(A \cap B \cap C)=\{6, 10, 11\}}$

Before giving the solution let discuss what is the universal set, Union, and intersection:

• A universal set (typically represented by U) is a collection that has items from all the sets, with no elements being repeated.
• It specifies in general as a set of all items to be considered.
• It describes all the elements or objects, including the elements, of other collections.
• The union of two sets is a collection with all A or B components (possibly both).
• The intersection denotes the resulting set between such two sets, which contains all the common items.

Following are the solution to the question:

Given:

Universal set $\bold{ S = \{ 1 , 2 , 3 , ... , 18 , 19 , 20 \} }$

Let

$\bold{A, B, C \subset S}$ i.e.

$\bold{A\subset S}\\\\\bold{B\subset S}\\\\\bold{C\subset S}$

Set $\bold{A = \{ 1 , 6 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 18 , 19 , 20\}}$

Set $\bold{B = \{ 1 , 3 , 4 , 6 , 8 , 9 , 10 , 11 , 12 , 15 , 20 \}}$

Set $\bold{C = \{ 3 , 6 , 7 , 8 , 9 , 10 , 11 , 13 , 14 , 17 \}}$

Find:

$\to \bold{(A \cup B\cup C) \cap(A\cup B\cup C) =?}\\\\\to \bold{(A \cap B\cap C) \cap(A \cap B \cap C)=?}$

Solution:

In $\bold{(A \cup B\cup C) \cap(A\cup B\cup C)}:$

Calculating:

$\bold{(A \cup B\cup C) }= \bold{\{1,3,4,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20 \}}$

In this question, the "$(A \cup B\cup C)$" value is equal to "$\bold{\{1,3,4,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20 \}}$" that is 18 elements.

When we intersecting both sets that is$\bold{(A \cup B\cup C) \cap(A\cup B\cup C)}$ so, its value will be the same that is:

"$\bold{\{1,3,4,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20 \}}$" .

In $\bold{(A \cap B\cap C) \cap(A \cap B \cap C):}$

Calculating:

$\bold{(A \cap B\cap C) }= \bold{\{6, 10, 11, \} }$

In this question, the "$\bold{(A \cap B\cap C) }$" value is equal to "$\bold{\{6, 10, 11 \} }$" that is 3 elements.

When we intersecting both sets that are $\bold{(A \cap B\cap C) \cap(A \cap B \cap C):}$ so, its value will be the same that is:

"$\bold{\{6, 10, 11 \} }$".

So, the final answer is:$\to \bold{(A \cup B\cup C) \cap(A\cup B\cup C) = \bold{\{1,3,4,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20 \}} }\\\\\to \bold{(A \cap B\cap C) \cap(A \cap B \cap C)=\{6, 10, 11\}}$Learn more:

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