Statistics Union Intersection How to Read It
The Union and Intersection of Ii Sets
- Folio ID
- 4738
Learning Outcomes
- Detect the wedlock of two sets.
- Find the intersection of two sets.
- Combine unions intersections and complements.
All statistics classes include questions about probabilities involving the matrimony and intersections of sets. In English, we employ the words "Or", and "And" to describe these concepts. For example, "Find the probability that a pupil is taking a mathematics class or a scientific discipline class." That is expressing the union of the two sets in words. "What is the probability that a nurse has a available's degree and more than five years of feel working in a hospital." That is expressing the intersection of 2 sets. In this section nosotros will learn how to decipher these types of sentences and volition learn about the significant of unions and intersections.
Unions
An element is in the union of two sets if information technology is in the start gear up, the second fix, or both. The symbol we utilise for the wedlock is \(\cup\). The word that you volition often encounter that indicates a wedlock is "or".
Instance \(\PageIndex{ane}\): Wedlock of 2 sets
Let:
\[A=\left\{2,v,vii,8\right\} \nonumber\]
and
\[B=\lbrace1,4,5,7,9\rbrace \nonumber \]
Find \(A\loving cup B\)
Solution
We include in the wedlock every number that is in A or is in B:
\[A\loving cup B=\left\{1,2,4,5,seven,8,9\right\} \nonumber \]
Example \(\PageIndex{2}\): Marriage of Two sets
Consider the following sentence, "Notice the probability that a household has fewer than 6 windows or has a dozen windows." Write this in set note as the wedlock of two sets so write out this union.
Solution
First, let A be the set of the number of windows that represents "fewer than 6 windows". This set includes all the numbers from 0 through five:
\[A=\left\{0,1,2,3,4,5\right\} \nonumber \]
Adjacent, allow B exist the gear up of the number of windows that represents "has a dozen windows". This is just the set that contains the single number 12:
\[B=\left\{12\right\} \nonumber \]
We can now observe the union of these two sets:
\[A\cup B=\left\{0,1,2,3,4,5,12\right\} \nonumber \]
Intersections
An element is in the intersection of ii sets if it is in the beginning prepare and it is in the second set. The symbol we employ for the intersection is \(\cap\). The word that you volition often see that indicates an intersection is "and".
Example \(\PageIndex{3}\): Intersection of Two sets
Allow:
\[A=\left\{3,iv,5,viii,9,ten,xi,12\right\} \nonumber \]
and
\[B=\lbrace5,6,7,eight,9\rbrace \nonumber \]
Notice \(A\cap B\).
Solution
We only include in the intersection that numbers that are in both A and B:
\[A\cap B=\left\{5,eight,nine\right\} \nonumber \]
Example \(\PageIndex{4}\): Intersection of 2 sets
Consider the following judgement, "Detect the probability that the number of units that a student is taking is more 12 units and less than 18 units." Assuming that students only accept a whole number of units, write this in set notation as the intersection of two sets and and so write out this intersection.
Solution
First, let A be the set of numbers of units that represents "more than 12 units". This set up includes all the numbers starting at xiii and continuing forever:
\[A=\left\{thirteen,\:14,\:15,\:...\right\} \nonumber \]
Side by side, let B exist the set of the number of units that represents "less than 18 units". This is the set that contains the numbers from one through 17:
\[B=\left\{1,\:2,\:3,\:...,\:17\right\} \nonumber \]
We can now find the intersection of these ii sets:
\[A\cap B=\left\{thirteen,\:14,\:15,\:16,\:17\right\} \nonumber \]
Combining Unions, Intersections, and Complements
One of the biggest challenges in statistics is deciphering a sentence and turning it into symbols. This tin be specially difficult when there is a judgement that does not have the words "union", "intersection", or "complement", but it does implicitly refer to these words. The all-time manner to become proficient in this skill is to practice, practice, and practice more.
Example \(\PageIndex{5}\)
Consider the following sentence, "If you ringlet a half-dozen sided dice, find the probability that it is not even and it is not a 3." Write this in gear up notation.
Solution
First, let A be the prepare of even numbers and B exist the set that contains just iii. We tin write:
\[A=\left\{2,four,half dozen\right\},\:\:\:B\:=\:\left\{three\right\} \nonumber \]
Next, since we want "non even" we need to consider the complement of A:
\[A^c=\left\{1,3,five\right\} \nonumber \]
Similarly since we desire "not a three", nosotros need to consider the complement of B:
\[B^c=\left\{i,two,4,5,6\right\} \nonumber \]
Finally, nosotros notice the primal word "and". Thus, we are asked to find:
\[A^c\cap B^c=\:\left\{1,3,5\right\}\cap\left\{one,2,4,5,vi\right\}=\left\{1,5\right\} \nonumber \]
Example \(\PageIndex{6}\)
Consider the following judgement, "If yous randomly select a person, find the probability that the person is older than 8 or is both younger than 6 and is not younger than 3." Write this in set notation.
Solution
First, let A be the set of people older than viii, B be the set of people younger than 6, and C be the fix of people younger than 3. We can write:
\[A=\left\{x\mid x>viii\right\},\:\:\:B\:=\:\left\{x\mid x<6\right\},\:C=\left\{x\mid x<3\right\} \nonumber \]
We are asked to find
\[A\cup\left(B\cap C^c\right) \nonumber \]
Notice that the complement of "\(< \)" is "\(\ge\)". Thus:
\[C^c=\left\{x\mid ten\ge3\right\} \nonumber \]
Next nosotros notice:
\[B\cap C^c=\left\{x\mid x<6\correct\}\cap\left\{10\mid x\ge3\right\}=\left\{x\mid3\le ten<6\right\} \nonumber \]
Finally, we find:
\[A\loving cup\left(B\cap C^c\right)=\:\left\{x\mid x>8\right\}\cup\left\{x\mid3\le x<6\right\} \nonumber \]
The clearest way to display this union is on a number line. The number line beneath displays the answer:
Exercise
Suppose that we pick a person at random and are interested in finding the probability that the person's birth month came after July and did not come up afterwards September. Write this event using ready notation.
- Ex: Find the Intersection of a Fix and A Complement Using a Venn Diagram
- Intersection and Complements of Sets
Source: https://stats.libretexts.org/Bookshelves/Introductory_Statistics/Support_Course_for_Elementary_Statistics/Sets/The_Union_and_Intersection_of_Two_Sets
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