Here you will learn what is the complement of a set definition with venn diagram and examples.
Letโs begin โ
Complement of a Set
Definition : Let U be the universal set and let A be a set such that A \(\subset\) U. Then, the complement of A with respect to U is denoted by Aโ or \(A^c\) or U โ A and is defined the set of all those elements of U which are not in A.
Thus, Aโ = {x \(\in\) U : x \(\notin\) A}
Clearly,ย x \(\in\) Aโย \(\iff\)ย x \(\in\) A.
Venn Diagram :
Also Read : Formulas and Operation of Sets
Example 1 : Let the set of natural numbers N = {1, 2, 3, 4, โฆ.. } be the universal set and let A = {2, 4, 6, 8, โฆ.}. Then Aโ = {1, 3, 5, โฆ.}/
Example 2 : If U = {1, 2, 3, 4, 5, 6, 7, 8, 9} and A = {1, 3, 5, 7, 9}, then Aโ = {2, 4, 6, 8}.
Following results are direct consequence of the definition of the complement of a set.
(i)ย Uโ = {x \(\in\) U : x \(\notin\) U} = \(\phi\)
(ii)ย \({\phi}โ\) = {x \(\in\) U : x \(\notin\) \(\phi\)} = U
(iii)ย (Aโ)โ = {x \(\in\) U : x \(\notin\) Aโ} = {x \(\in\) U : x \(\in\) A} = A
(iv)ย A \(\cup\) Aโ = U
(v)ย A \(\cap\) Aโ = \(\phi\)
