Venn Diagrams and Set Notation

A Venn Diagram is a way to visualize set relations between a finite number of sets. Below is a Venn Diagram for three sets \(T, D,\) and \(H\).

Venn Diagram Sets

Venn Diagram Sets

We introduce some notation from Set Theory:

  1. \( |T|\) is the number of elements in set \( T\).

  2. Intersection of two sets, denoted \( \cap\), refers to the elements that are in both sets. In the example, \( T \cap D = \{ d, g\} \).

  3. Union of two sets, denoted \( \cup\), refers to the elements that are in at least one of the two sets. In the example, \( T \cup H = \{a, c, d, e, f, g\} \).

  4. Complement (Absolute), denoted \( ^c\), refers to the elements that are not in the set. In the example, \( D^c = \{ a, c, e, i\} \).

  5. Complement (Relative), denoted \( \backslash\), refers to the elements in the first set, but are not in the second set. In the example, \( H\backslash T = \{ c, f \} \).

  6. Symmetric Difference, denoted \( \triangle\), refers to the elements that are in at least one of the two sets, but are not in both sets. In the example, \( D \triangle H = \{b, c, d, e\} \).

Note by Arron Kau
4 years, 8 months ago

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H\T can also be written H - T.

Bhagirath Mehta - 4 years, 4 months ago

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Ya ......dear right

Vishal Bambhaniya - 3 years, 10 months ago

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As a non-mathematician I found some of the thinking in these examples quite puzzling and hard to follow!

David Chandler - 1 year, 4 months ago

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Complement of H = H'

Arijit Konar - 2 years, 6 months ago

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this is kinda like logic gates

Laura Gao - 9 months, 1 week ago

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Enlightening

Williams Ezegrim - 1 year ago

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how do you solve when there is an unknown is a given set?

Anthony Ishmael - 4 years, 5 months ago

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You need to turn the rules into English and use logic. Like in the question that you probably came from, it says that: 0 is in the set. If \(p\) is in the set, and \(q\) is in the set, and \(p \neq q\), then \(p+q\) is in the set. Now, if 10 is in the set, then -10 being in the set wouldn't break any rules, becuase \(-10 \neq 10\) and \(-10+10=0\) which is also in the set.

Set notation isn't useful in problem solving, but it helps show the question in a short way. Imagine if we didn't have any math symbols. Then \(3x-9= 0, x=3\) would be: "Take a number. If you take away nine from 3 times that number the answer is zero. The number is 3." Which is easier to understand?

Samuel Godswill - 5 months ago

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yeah,, idont find this post useful in solving,, please how do you actually solve???

Louie Arnold Panganiban - 4 years, 4 months ago

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