View source for Complement of a Set

= Complement of a Set - Definition, Explanation, and Examples =

The '''complement''' of a set contains all elements that are not in the set but belong to a larger, universal set. It helps identify what is "outside" a given set within a specified context.

== Definition of Complement ==

Let <math>U</math> be the universal set, which contains all elements under consideration. The complement of a set <math>A</math>, denoted by <math>A'</math> or <math>\overline{A}</math>, is defined as:

<math>
A' = \{ x \in U : x \notin A \}
</math>

In words, the complement of <math>A</math> is the set of all elements in the universal set <math>U</math> that are not in <math>A</math>.

== Understanding Complement ==

The complement tells us everything outside the set <math>A</math> within the universe of discourse.

== Step-by-Step Explanation ==

1. Identify the universal set <math>U</math>.  
2. Identify the elements of the set <math>A</math>.  
3. Find all elements in <math>U</math> that are not in <math>A</math>.  
4. Collect these elements to form the complement set <math>A'</math>.

== Examples of Complement of a Set ==

=== Example 1: Numbers ===  
Let the universal set be:

<math>U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}</math>

and let

<math>A = \{2, 4, 6, 8, 10\}</math>

Step 1: Universal set <math>U</math> contains numbers 1 to 10.  
Step 2: Set <math>A</math> contains even numbers 2, 4, 6, 8, 10.  
Step 3: Elements in <math>U</math> but not in <math>A</math> are odd numbers: 1, 3, 5, 7, 9.  
Step 4: Complement of <math>A</math> is:

<math>A' = \{1, 3, 5, 7, 9\}</math>

=== Example 2: Letters ===  
Let

<math>U = \{\text{a}, \text{b}, \text{c}, \text{d}, \text{e}\}</math>

and

<math>B = \{\text{a}, \text{c}, \text{e}\}</math>

Then,

<math>B' = \{\text{b}, \text{d}\}</math>

since these are the letters in <math>U</math> not in <math>B</math>.

=== Example 3: Shapes ===  
Consider a universal set of shapes:

<math>U = \{\text{circle}, \text{square}, \text{triangle}, \text{rectangle}\}</math>

and set

<math>C = \{\text{circle}, \text{triangle}\}</math>

The complement of <math>C</math> is:

<math>C' = \{\text{square}, \text{rectangle}\}</math>

=== Example 4: Numbers Between 1 and 15 ===  
Let

<math>U = \{1, 2, 3, \dots, 15\}</math>

and

<math>D = \{5, 6, 7, 8, 9\}</math>

The complement of <math>D</math> is all numbers from 1 to 15 except 5 through 9:

<math>D' = \{1, 2, 3, 4, 10, 11, 12, 13, 14, 15\}</math>

=== Example 5: Prime Numbers Up to 20 ===  
Let

<math>U = \{1, 2, 3, \dots, 20\}</math>

and

<math>E = \{2, 3, 5, 7, 11, 13, 17, 19\}</math>  (the prime numbers)

Then the complement of <math>E</math> is all numbers from 1 to 20 that are not prime:

<math>E' = \{1, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20\}</math>

== Important Notes ==

* The complement depends on the universal set <math>U</math>.  
* The union of a set and its complement is the universal set: <math>A \cup A' = U</math>.  
* The intersection of a set and its complement is the empty set: <math>A \cap A' = \emptyset</math>.

== Summary ==

The complement of a set shows all elements outside the set within a defined universal set. It is a useful concept for understanding what is excluded from a set.

[[Category:Set Theory]]
[[Category:Set Operations]]
[[Category:Mathematics]]