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Chapter 2: Problem 56

If \(A \cap B=\varnothing\), then \(n(A \cup B)=n(A)+n(B)\).

Short Answer

Expert verified
The number of elements in the union of two sets with no common elements is equal to the sum of the number of elements in each set individually, i.e., if \(A \cap B=\varnothing\), then \(n(A \cup B)=n(A)+n(B)\). The proof involves understanding of the union (\(A \cup B\)) and intersection (\(A \cap B\)) of sets and the notion of 'double-counting' of elements.

Step by step solution

01

Understand The Term

The statement means that when two sets \(A\) and \(B\) have no common elements (the intersection is an empty set), the count of elements (\(n\)) in \(A \cup B\) (union of \(A\) and \(B\)) is equal to the sum of the count of elements in set \(A\) and set \(B\) separately (\(n(A)+n(B)\)). This is shown as \(A \cap B=\varnothing\) then \(n(A \cup B)=n(A)+n(B)\).
02

Define Union and Intersection

In set theory, the union of two sets (\(A \cup B\)) is a set that contains all elements that are either in \(A\) or in \(B\) or in both. The intersection of two sets (\(A \cap B\)) is a set that contains all elements that are both in \(A\) and \(B\). If \(A \cap B=\varnothing\), this means that \(A\) and \(B\) share no elements.
03

Analyze the Statement

If \(A \cap B=\varnothing\), then the elements of \(A \cup B\) will simply be the separate elements of \(A\) and \(B\) because there are no shared elements to account for. Therefore, the total count will be equal to the sum of the count of elements in \(A\) and \(B\), expressed as \(n(A \cup B)=n(A)+n(B)\).
04

Prove the Statement

Assuming \(A \cap B=\varnothing\), meaning \(A\) and \(B\) have no common elements. Therefore, every element in \(A \cup B\) is either in \(A\) or in \(B\) but not in both. There isn't any 'double-counting' of elements causing the total count in the union to be \(n(A)+n(B)\). Therefore, \(n(A \cup B)=n(A)+n(B)\). This concludes the proof.

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Most popular questions from this chapter

In Exercises 41-66, let $$ \begin{aligned} U &=\\{\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}, \mathrm{f}, \mathrm{g}, \mathrm{h}\\} \\ A &=\\{\mathrm{a}, \mathrm{g}, \mathrm{h}\\} \\ B &=\\{\mathrm{b}, \mathrm{g}, \mathrm{h}\\} \\ C &=\\{\mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}, \mathrm{f}\\} \end{aligned} $$ Find each of the following sets. \(B^{\prime} \cap C\)

Let $$ \begin{aligned} U &=\\{\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}, \mathrm{f}, \mathrm{g}, \mathrm{h}\\} \\ A &=\\{\mathrm{a}, \mathrm{g}, \mathrm{h}\\} \\ B &=\\{\mathrm{b}, \mathrm{g}, \mathrm{h}\\} \\ C &=\\{\mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}, \mathrm{f}\\} \end{aligned} $$ Find each of the following sets. \(\left(A^{\prime} \cap B\right) \cup\left(A^{\prime} \cap C^{\prime}\right)\)

In Exercises 41-66, let $$ \begin{aligned} U &=\\{\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}, \mathrm{f}, \mathrm{g}, \mathrm{h}\\} \\ A &=\\{\mathrm{a}, \mathrm{g}, \mathrm{h}\\} \\ B &=\\{\mathrm{b}, \mathrm{g}, \mathrm{h}\\} \\ C &=\\{\mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}, \mathrm{f}\\} \end{aligned} $$ Find each of the following sets. \(A \cup U\)

In Exercises 41-66, let $$ \begin{aligned} U &=\\{\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}, \mathrm{f}, \mathrm{g}, \mathrm{h}\\} \\ A &=\\{\mathrm{a}, \mathrm{g}, \mathrm{h}\\} \\ B &=\\{\mathrm{b}, \mathrm{g}, \mathrm{h}\\} \\ C &=\\{\mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}, \mathrm{f}\\} \end{aligned} $$ Find each of the following sets. \(B \cup U\)

In Exercises 41-66, let $$ \begin{aligned} U &=\\{\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}, \mathrm{f}, \mathrm{g}, \mathrm{h}\\} \\ A &=\\{\mathrm{a}, \mathrm{g}, \mathrm{h}\\} \\ B &=\\{\mathrm{b}, \mathrm{g}, \mathrm{h}\\} \\ C &=\\{\mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}, \mathrm{f}\\} \end{aligned} $$ Find each of the following sets. \((A \cap B)^{\prime}\)
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