De Morgan's First Law

De Morgan’s Laws:

A well-defined collection of objects or elements is known as a set. Various operations like complement of a set, union and intersection can be performed on two sets. These operations and their usage can be further simplified using a set of laws known as De Morgan’s Laws. These are very easy and simple laws.

Any set consisting of all the objects or elements related to a particular context is defined as a universal set. Consider a universal set U such that A and B are the subsets of this universal set.

According to the De Morgan’s first law, the complement of union of two sets A and B is equal to the intersection of the complement of the sets A and B.

(A∪B)’= A’∩ B’      —–(1)

Where complement of a set is defined as

A’= {x:x ∈ U and x ∉ A}

Where A’ denotes the complement.

This law can be easily visualized using Venn Diagrams.

The L.H.S of the equation 1 represents the complement of union of two sets A and B. First of all, union of two setsA and B is defined as the set of all elements which lie eitherin set A or in set B. It can be visualized using Venn Diagrams as shown:

De Morgan's theorem

Figure 1 Union of Sets

The highlighted or the green colored portion denotes A∪B. The complement of union of A and B i.e., (A∪B)’is set of all those elements which are not in A∪B. This can be visualized as follows:

de morgan's law proof

Figure 2 Complement of sets

Similarly, R.H.S of equation 1 can be represented using Venn Diagrams as well, the first part i.e., A’ can be depicted as following:

de morgan's law set theory

Figure 3 Complement of set A

The portion in black indicates set A and blue part denotes its complement i.e., A’.
Similarly, B’ is represented as:

de morgan's law definition

Figure 4 Complement of set B

The portion in black indicates set B and yellow part denotes its complement i.e., B’.

If fig. 3 and 4 are superimposed on one another, we get the figure similar to that of complement of sets.

de morgan's law

Figure 5 Intersection of complements of sets

Hence L.H.S = R.H.S


As, A∪B= either in A or in B

(A∪B)’= L.H.S = neither in A nor in B

Also, A’= Not in A

B’= Not in B

A’∩ B’= Not in A and not in B

⇒(A∪B)’= A’∩ B’

Thus, by visualizing the Venn Diagrams and analyzing De Morgan’s Laws by writing it down, its validity can be justified.

Practise This Question

Anita, Sunita, Ram and Raman scored 89, 87, 90 and 55 marks (out of 100) respectively in an exam. Who scored the maximum percentage and how much?