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14.4 Union and intersection

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14.3 Venn diagrams
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14.5 Probability identities

14.4 Union and intersection (EMA7Z)

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Union

The union of two sets is a new set that contains all of the elements that are in at least one of the two sets. The union is written as \(A \cup B\) or “\(A \text{ or } B\)”.

Intersection

The intersection of two sets is a new set that contains all of the elements that are in both sets. The intersection is written as \(A \cap B\) or “\(A \text{ and } B\)”.

The figure below shows the union and intersection for different configurations of two events in a sample space, using Venn diagrams.

ffa836033f01e5c5657502b3eba43f4e.png 425af81fda2eebd0900276cd0611b963.png 6e450064da0f95f67b5b6381d6e02a2a.png
e9e323af04473efebfe0fef9417543e9.png 8b8db10b13d9cfbbd3d28dcbe5b46c9c.png f0741883bab68a398322e922887bdf92.png
177bef6979699d04dfbd5aecc23c31d5.png ba17ac4beeb710d9f4b00edc65f7d916.png 3dedb7c96bd626920bccadc876c2d053.png

Figure 14.1: The unions and intersections of different events. Note that in the middle column the intersection, \(A \cap B\), is empty since the two sets do not overlap. In the final column the union, \(A \cup B\), is equal to \(A\) and the intersection, \(A \cap B\), is equal to \(B\) since \(B\) is fully contained in \(A\).

Textbook Exercise 14.4

A group of learners are given the following Venn diagram:

4bd17fae44bbde95773717adebeae4e2.png

The sample space can be described as \(\{ n:n \text{ } \epsilon \text{ } \mathbb{Z}, \text{ } 1 \leq n \leq 15 \}\).

They are asked to identify the event set of the intersection between event set \(A\) and event set \(B\), also written as \(A \cap B\). They get stuck, and you offer to help them find it.

Which set best describes the event set of \(A \cap B\)?

  • \(\{7;10;11\}\)
  • \(\{1;2;3;4;5;6;7;9;10;11\}\)
  • \(\{1;2;3;4;5;6;7;9;10\}\)
  • \(\{7;10\}\)

The intersection between event set \(A\) and event set \(B\), also written as \(A \cap B\), can be shaded as follows:

4fbc995dd5869d8e99e692ceb8270878.png

Therefore the event set \(\{7;10\}\) best describes the event set of \(A \cap B\).

A group of learners are given the following Venn diagram:

376d27da23fff90f63f2e37aaff4dffa.png

The sample space can be described as \(\{ n:n \text{ } \epsilon \text{ } \mathbb{Z}, \text{ } 1 \leq n \leq 15 \}\)

They are asked to identify the event set of the union between event set \(A\) and event set \(B\), also written as \(A \cup B\). They get stuck, and you offer to help them find it.

Which set best describes the event set of \(A \cup B\)?

  • \(\{1;6;7;10;15\}\)
  • \(\{1;2;4;5;6;7;8;9;10;11;12;13;14;15\}\)
  • \(\{2;4;5;9;10;11;12;13;14\}\)
  • \(\{3\}\)

The union between event set \(A\) and event set \(B\), also written as \(A \cup B\), can be shaded as follows:

5dc6b34374c8568237ca826ca9516a33.png

Therefore the event set \(\{1;2;4;5;6;7;8;9;10;11;12;13;14;15\}\) best describes the event set of \(A \cup B\).

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14.3 Venn diagrams
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14.5 Probability identities