De Morgan's First Law

De Morgon’s Law states that the complement of the union of two sets is the intersection of their complements and the complement of the intersection of two sets is the union of their complements. These are mentioned after the great mathematician De Morgan. This law can be expressed as ( A ∪ B) ‘ = A ‘ ∩ B ‘. In set theory, these laws relate the intersection and union of sets by complements.

De Morgan’s Laws Statement and Proof

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 De Morgan’s first law, the complement of the 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:

Union of Sets

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:

Complement of sets

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 follows:

Complement of set A

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:

 Complement of set B

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 the complement of sets.

 Intersection of complements of sets

Figure 5 Intersection of complements of sets

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

Mathematically,

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.

2 Comments

  1. Good graphic method

  2. Hello Sir
    Please answer these questions for me i really need it right now.
    Question 1: Prove the DeMorgan law A={1,2,3,4), B=(3,4,5,6}?
    Question 2: Find the Power set of A={0,1,2,3,4,5,6}
    with full written answer

    Thank You

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