Set theory defines a complement of a set as the set of elements not included in A. Considering all sets in the universe to be members of a given set U, the absolute complement of A is the set of elements in U that are not part of A. It is also called the set difference between B and A, which is the set of elements in B that are not in A in different Types of Sets.
It is the set that includes all the elements of the universal set that are not included in the given set. A is a set of all coins, which is a subset of a universal set that includes all coins and notes. Set A’s complement is a set of notes (which does not include coins). Our article discusses the complement of a set in detail, along with its definition, properties, and examples.
What is the Complement of a Set?
In the case of a universal set (U) containing a subset A, the complement of set A is represented as A’, and includes the elements of the universal set but not the elements of set A. Here, A’ = {x ∈ U : x ∉ A}. A set’s complement is its difference from a universal set.
Complement of Set Symbol
A’, B’, C’, etc., are the complements of any set. Alternatively, we can say that if the universal set (U) and the subset of the universal set (A) are given, then the difference between the universal set and the subset is the complement of the subset, that is A’ = U – A.
Example of Complement of a Set
A complement of set A is other than its elements if the universal set is all prime numbers up to 25 and set A = [2, 3, 5].
The first step is to check for the universal set and the set for which the complement must be found. U = {2, 3, 5, 7, 11, 13, 17, 19, 23}, A = {2, 3, 5}.
The second step is to subtract (U – A). Here,
U – A = A’
= {2, 3, 5, 7, 11, 13, 17, 19, 23} – {2, 3, 5}
= {7, 11, 13, 17, 19, 23}
Set Theory
Let’s work with objects instead of numbers. There are a variety of types of collections, such as a group of colors, a group of food items, a group of garments, a group of books, etc.
When mathematically represented, this collection or group of objects is called a set.
The elements of a set are called elements or entities, and the sets are often shown in curly brackets.
A ‘Theory of sets’ or ‘Set Theory’ was created by Georg Cantor, a German mathematician and logician.
Let’s examine the definition of set theory, the basic rules of set theory, the types of sets, the symbols of set theory.
Set Theory Definition
Objects called sets are grouped together in set theory, which is a branch of mathematics.
Consider the following example to understand set theory: we will list all the colors of the rainbow.
The colors violet, indigo, blue, green, yellow, orange, and red.
There is a good definition of colors in the above-mentioned list. In this list, fruit and flower names are not allowed. Known as a set, such a list of elements is well-defined.
When the same list is reversed
A variety of reds, oranges, yellows, greens, blue indigos, and violets.
Since sets do not care about the order of their elements, the set does not alter.
Sets are written using curly braces and capitalized.
