Proofs for De Morgan's law





Proofs for De Morgan's law can be easily done by using Venn diagram.

The De Morgan's laws are:

      (i)(A∪B)'=A'∩B'.

      (ii)(A∩B)'=A'∪B'.

     
Here we have to use the universal set, as we are going to deal with complement and difference. To represent universal set we have to use rectangle.

(i) (A∪B)'=A'∩B'

 Here we will take the left hand side and first draw the diagram for the union of A and B.  After that we will do the complement for AuB.

Left hand side:

Right hand side: Here first we draw the diagrams for complement of A and B. After that we will do the union of both those sets.

So last diagram of both left hand side and right hand side are same. So we have proved (A∪B)'=A'∩B'.


 (ii)(A∩B)'=A'∪B'

 First we will take the left hand side. In that we will draw diagram for AnB. After that we will find the complement. Secondly we will take the right hand side. Here we will first find the complements of A and B. After that we will find the union of both the sets A' and B'.

Left hand side:

Right hand side:

The last diagram of both left hand and right hand side are same. Hence we proved (A∩B)'=A'∪B'.

 We have proved De Morgan's laws.

The following two laws are also said to be De Morgan's laws.

                   A-(B∪C)=(A-B)∩(A-C).

                   A-(B∩C)=(A-B)∪(A-C)

Let us see the proof of these laws.

(i) A-(B∪C)=(A-B)∩(A-C)

  We will take the left hand side and draw the diagram for (B∪C) and after that we will draw the diagram for A-(B∪C). Next we will take the right hand side. Here we will first draw the diagrams for A-B and A-c. After that we will find the intersection of both the sets.

Left hand side: 

Right hand side:

The last diagram for both left hand side and right hand side are same. Thus

we have proved A-(B∪C)=(A-B)∩(A-C).

Now let us prove

                         A-(B∩C)=(A-B)∪(A-C).

Here we will take the left hand side first, and draw BnC. After that we will draw the diagram for A-(BnC). Next we will take the right hand side and draw the diagrams for A-B and A-C. After that we will find the union of both the sets.

Left hand side:

Right hand side:

Here the last diagram of both left hand side and right hand side are same.

Thus we have proved A-(B∩C)=(A-B)∪(A-C).

Students practice all the proofs. If you have any doubt please contact us using contact us page. We will Clear all your doubts.





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