SET LANGUAGE

Operations on Sets
1. Union of sets : A U B = {x / x є A or x є B or x є A and x є B }
Note:
A U A = A
A n A = A
A n A’= {}
A U A’= U
2. Intersection of sets: A n B = {x / x є A and x є B }
Properties of set difference
1. Set difference is not commutative A – B ≠ B – A
2. Set difference is not associative A– (B – C) ≠ (A – B) – C
Distributive Property
1. Union is distributed over intersection A U (B n C) = (A U B ) n (AU C)
2. Intersection is distributed over union A n ( B U C) = (A n B) U (A n C)
DE Morgan’s Laws
1. Regarding complementation
(i) (A U B)’ = A’ n B’
(ii) (A n B)’ = A’ U B’
2. Regarding set difference
(i) A – (B U C) = (A – B) n (A – C)
(ii) A – (B n C) = (A – B) U (A – C)
♦ n(A U B) = n (A) + n (B) – n(A n B) if A n B ≠ { }
♦ n(A U B) = n (A) + n (B) if A n B = { }

• n(A U B U C) = n (A) + n (B) + n (C)–n(A n B)–n(B n C)–n(A n C) + n(A n B n C)

Mr.Ramanujan

The Great Indian Mathematician.

• Srīnivāsa Rāmānujan FRS (Tamil: ஸ்ரீநிவாச ராமானுஜன்) (22 December 1887 – 26 April 1920) was an Indian mathematician. With almost no formal training in pure mathematics, he made substantial contributions in the areas of mathematical analysis, number theory, infinite series and continued fractions.
• Born and raised in Erode, Tamil Nadu, India, Ramanujan first encountered formal mathematics at age ten. He demonstrated a natural ability, and was given books on advanced trigonometry by S. L. Loney.He had mastered them by age thirteen, and even discovered theorems of his own.
• He demonstrated unusual mathematical skills at school, winning accolades and awards. By the age of seventeen, he was conducting his own mathematical research on Bernoulli numbers and the Euler–Mascheroni constant. He received a scholarship to study at Government College in Kumbakonam, but lost it when he failed his non-mathematical coursework. He joined another college to pursue independent mathematical research, working as a clerk in the Accountant-General's office at the Madras Port Trust Office to support himself.
• In 1912-1913, he sent samples of his theorems to three academics at the University of Cambridge. Only G. H. Hardy recognized the brilliance of his work, and he asked Ramanujan to study under him at Cambridge.
  Ramanujan independently compiled nearly 3900 results (mostly identities and equations) during his short lifetime.
• Although a small number of these results were actually false and some were already known, most of his claims have now been proven to be correct.He stated results that were both original and highly unconventional, such as the Ramanujan prime and the Ramanujan theta function, and these have inspired a vast amount of further research.
• However, some of his major discoveries have been rather slow to enter the mathematical mainstream. Recently, Ramanujan's formulae have found applications in the field of crystallography and in string theory. The Ramanujan Journal, an international publication, was launched to publish work in all the areas of mathematics that were influenced by his work.