Complement of a set

Complement of a Set Definition

The Complement of a set A may be defined as the set of all the elements that are present in its Universal Set U but not in the set A.

To understand the concept of Complement of a Set, let us consider a set A=\{1, 2, 3, 4, 5\} and let the Universal Set of A be U=\{1, 2, 3, 4, 5, 6, 7, 8, 9\}.

The Complement of Set A will have all the elements that are present in the Universal Set, U but not in A. In this case, 6, 7, 8 and 9 are such elements. Therefore, the Complement of Set A will have the elements 6, 7, 8 and 9.

Set Builder Notation for Complement of a Set

The Set Builder Notation for the Complement of a Set is given by,

\displaystyle {A}'=[{x:x\in U,x\notin A}]

Complement of a set Symbol

The Symbol for representing the Complement of a Set A is given by {{A}'} or sometimes also given by {{A}^{C}}.

Complement of a Set Examples

Let us consider a set A = \{1, 2, 3\} and its universal set B is

B = \{1, 2, 3, 4, 5, 6, 7\}

Complement of a set venn diagram

The above figure shows the Venn diagram of the Complement of a Set. Here, the Universal Set B has some extra elements i.e, 4, 5, 6 and 7 that are not present in the set A, shown by the green shade in the above figure. The set of these elements are the Complement of the Set A.

Therefore,

\displaystyle {{A}'} = \{4, 5, 6, 7\}

Properties of Complement of a Set

  • The Union of set A with its Complement \displaystyle {{A}'} will give the universal set.

\displaystyle A\cup {A}'=U

  • The Intersection of set A with its Complement \displaystyle {{A}'} will result an Empty Set.

\displaystyle A\cap {A}'=\phi

  • The Complement of an Empty set \displaystyle \phi is the universal set U.

\displaystyle {\phi }'=U

  • The Complement of an Universal set U is an Empty set.

\displaystyle {U}'=\phi

  • If \displaystyle A\subseteq B then \displaystyle {B}'\subseteq {A}'

\displaystyle ({A}'{)}'=A

How to Find the Complement of a Set?

Step 1: Let A=\{1, 2, 3\}

Step 2: Let the universal set of A is U=\{1, 2, 3, 4, 5, 6\}

Step 3: Find the elements that are present in the universal set U but not in A.

The elements are: \{4, 5, 6\}

Step 4: The above elements found in Step 3 are the complement of set A.

Complement of A, {A}' = \{4, 5, 6\}

Frequently Asked Questions (FAQ)

The Complement of a set A may be defined as the set of all the elements that are present in its Universal Set U but not in set A.

Step 1: Let A={1, 2, 3}

Step 2: Let the universal set of A is U={1, 2, 3, 4, 5, 6}

Step 3: Find the elements that are present in the universal set U but not in A.

The elements are: {4, 5, 6}

Step 4: The above elements found in Step 3 are the complement of set A.

Complement of A, A' = {4, 5, 6}

The complement of a set is the collection of all the elements that are present in its universal set but not in the set.

The Properties of Complement of a Set are:

  1. The union of set A with its complement A' will give the universal set.
    2. The intersection of set A with its complement A' will result in an Empty Set.
    3. The complement of an Empty set phi is the universal set U.
    4. The complement of a universal set U is an Empty set.
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