On this page
Most students meet De Morgan's laws as two lines to memorise in the Class 11 Sets chapter. Fewer know that the man they are named after was born in Tamil Nadu, or that the same two lines also run the logic gates in Class 12 electronics. This post covers his life, the laws themselves, worked examples, and the mistakes that cost marks.
Who was Augustus De Morgan?
De Morgan was born on 27 June 1806 in Madura, in the Madras Presidency, now Madurai in Tamil Nadu, according to his MacTutor biography. His father was an officer in the Madras Native Infantry. He lost the sight of one eye soon after birth, and the family returned to England when he was seven months old.
He entered Trinity College, Cambridge at 16 and was Fourth Wrangler in 1827. Because Cambridge then required a religious test for higher degrees and fellowships, which he refused to sign, he could go no further there. In 1827, aged 21 and with no published research, he applied for the new chair of mathematics at London University. On 23 February 1828 he became its first professor of mathematics. He resigned twice on matters of principle, in 1831 and again in 1866, having been reappointed in 1836.
MacTutor credits him with introducing De Morgan's laws, and notes that his greatest contribution was as a reformer of mathematical logic. In 1838 he defined and named "mathematical induction", the method you use in Class 11 proofs. In 1864 he co-founded the London Mathematical Society and became its first president. He also tutored Ada Lovelace.
What are De Morgan's laws?
The NCERT Class 11 Sets chapter states them for any two subsets A and B of a universal set U:
(A ∪ B)′ = A′ ∩ B′. The complement of a union is the intersection of the complements.
(A ∩ B)′ = A′ ∪ B′. The complement of an intersection is the union of the complements.
The chapter adds that they can be verified with Venn diagrams. The JEE (Main) 2026 syllabus lists union, intersection and complement of sets and their algebraic properties in Unit 1, so these laws are fair game.
Worked example 1: check both laws on a small set
Let U = {1, 2, 3, ..., 10}, A = even numbers = {2, 4, 6, 8, 10}, and B = multiples of 3 = {3, 6, 9}.
First law:
A ∪ B = {2, 3, 4, 6, 8, 9, 10}, so (A ∪ B)′ = {1, 5, 7}.
A′ = {1, 3, 5, 7, 9} and B′ = {1, 2, 4, 5, 7, 8, 10}.
A′ ∩ B′ = {1, 5, 7}. Both sides match.
Second law:
A ∩ B = {6}, so (A ∩ B)′ = {1, 2, 3, 4, 5, 7, 8, 9, 10}.
A′ ∪ B′ = {1, 2, 3, 4, 5, 7, 8, 9, 10}. Both sides match.
Worked example 2: counting with complements
Question. How many numbers from 1 to 100 are divisible by neither 2 nor 3?
"Neither 2 nor 3" means not in A and not in B, which is A′ ∩ B′. By De Morgan's first law that equals (A ∪ B)′, which is easier to count.
n(A) = 50 multiples of 2 and n(B) = 33 multiples of 3.
n(A ∩ B) = 16, the multiples of 6 up to 96.
n(A ∪ B) = 50 + 33 − 16 = 67.
n((A ∪ B)′) = 100 − 67 = 33.
So 33 numbers are divisible by neither.
Where else do De Morgan's laws appear?
The same rules hold in logic, with "or" for union, "and" for intersection and "not" for complement. "It is not true that it is raining or cold" means "it is not raining and it is not cold". In electronics, this is why a NOR gate gives the same output as an AND gate with both inputs inverted. Our post on semiconductors and logic gates shows truth tables you can check this against.
Common mistakes students make
Keeping the symbol the same. (A ∪ B)′ is not A′ ∪ B′. In the example above, A′ ∪ B′ has nine elements while (A ∪ B)′ has three. The complement flips ∪ to ∩ and back.
Forgetting the universal set. A′ only makes sense once U is fixed. Write U down first.
Subtracting the overlap twice, or not at all. n(A ∪ B) = n(A) + n(B) − n(A ∩ B). Draw the Venn diagram before you add.
Losing the brackets. A′ ∩ B′ and (A ∩ B)′ are different sets.
For a full walk-through of the chapter, see our guide to sets and relations in Class 11. If you want a clean explanation of complements or Venn diagrams before trying more questions, the Fermi Concept Library covers every concept in mathematics, each with its own lesson.
What can a student take from De Morgan's way of working?
Test a rule with your own numbers. In the preface to his Elements of Arithmetic (1830), as quoted by MacTutor, he told students they could convince themselves a rule was true by working each demonstration again with different numbers from those in the text. That is exactly what worked example 1 does.
Teach to learn. A description quoted by MacTutor says that once he began teaching others, he began to teach himself better than he had been taught. Explaining a law to a friend shows you whether you really have it.
Hold to your principles. He turned down a path at Cambridge and twice resigned his chair rather than accept what he thought was wrong.
Make your own U, A and B with ten elements and check both laws by hand before your next maths class.
