Set Complement

The complement of a set $S$ is the set of all elements that are not in $S.$

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, $H\backslash T = \{ c, f \}$.

Complement (Absolute), denoted $^c$, refers to the elements that are not in the set.

Complement (Relative), denoted $\backslash$, refers to the elements in the first set, but are not in the second set.

How many ways are there to rearrange the letters in the word "JAPAN" such that there are no A's together?

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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. $_\square$

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 $A^c = \{0, 2, 4, 6, 7, 8, 9 \} .$ $_\square$