Set Complement
The complement of a set \(S\) is the set of all elements that are not in \(S.\)
Contents
Complement - Definition
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
Complement (Absolute), denoted \( ^c\), refers to the elements that are not in the set. In the example, \( D^c = \{ a, c, e, i\} \).
Complement (Relative), denoted \( \backslash\), refers to the elements that are in the first set, but are not in the second set. In the example, \( H\backslash T = \{ c, f \} \).
Absolute Complement
Complement (Absolute), denoted \( ^c\), refers to the elements that are not in the set.
Relative Complement
Complement (Relative), denoted \( \backslash\), refers to the elements in the first set, but are not in the second set.
Complement - Problem Solving
How many ways are there to rearrange the letters in the word "JAPAN" such that there are no A's together?
Consider the complement, that is, we want to find the number of ways to rearrange the letters such that the A's are together. By permutation, we have \(4!=24\) ways.
The total number of possible permutation of the word in question is \(\frac{5!}{2!} = 60. \)
Taking their difference shows that there are 36 ways such that there are no A's together. \(_\square\)
How many arrangements are there of the letters of BANANA such that no two N's appear in adjacent positions?
A movie company wants to make movies featuring various monsters. They want each movie to have at least 2 different monsters and no two movies to have the exact same set of monsters. What is the minimum number of monsters the company must use in order to make a million movies?
Details and assumptions
1 million = \(1, 000 , 000 \).
The complement of a set \(A\) refers to all of the elements that are not in \(A\). Note that we have to define the universe \(U\) that \(A\) is contained within. The complement of \(A\) is denoted as \( A^c \).
The relative complement of set \(A\) with respect to set \(B\), is the set of elements that are in \(B\), but not in \(A\). This is denoted as \( B \text{ \ } A \). Using set notation, we can also denote this as \( B \cap A^c \).
Let \(D\) be the set of digits. If \(A = \{ 1, 3, 5 \}\) is a subset of \(D,\) what is \( A^c ?\)
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Since \( D = \{0,1,2,3,4,5,6,7,8,9 \} ,\) \( A^c\) is the set of elements that are not in \(A,\) which gives us \( A^c = \{0, 2, 4, 6, 7, 8, 9 \} .\) \(_\square\)