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The number of elements in a set is called its cardinal number and is written as n(A). A set with cardinal number 0 is called a null set while that with cardinal number ∞ is called an infinite set.
Set A is said to be a subset of Set B if each and every element of Set A is also contained in Set B. Set A is said to be a proper subset of Set B if Set B has at least one element that is not contained in Set A. A set with ‘n’ elements will have 2n subsets (2n – 1 proper subsets)
The Universal set is defined as the set of all possible objects under consideration.

EduRev's Tip: Any set is a subset of itself, but not a proper subset. The empty set, denoted by ∅ , is also a subset of any given set X. The empty set is always a proper subset, except of itself. Every other set is then a subset of the universal set.

Union of two sets is represented as A ∪ B and consists of elements that are present in either Set A or Set B or both. Intersection of two sets is represented as A ∩ B and consists of elements that are present in both Set A and Set B. n(A ∪ B) = n(A) + n(B) — n(A ∩ B)

Venn Diagram: A venn diagram is used to visually represent the relationship between various sets. What do each of the areas in the figure represent?
Important Formulae: Set Theory | CSAT Preparation - UPSC
I – only A; II – A and B but not C; III – Only B; IV – A and C but not B; V – A and B and C; VI – B and C but not A; VII – Only C
n(A ∪ B ∪ C) = n(A) + n(B) + n(C) — n(A ∩ B) — n(A ∩ C) -n(B ∩ C) + n(A ∩ B ∩ C)

Binomial Theorem

For some basic values:

  • (a + b)° = 1
  • (a + b)1 = a + b
  • (a + b)2 = a2 + 2ab + b2
  • (a + b)3 = a3 + 3a2b + 3ab2 + b3
  • (a + b)4 = a+ 4a3b + 6a2b2 + 4ab3 + b4
  • (a + b)5 = a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4 + b5

Theorem
(a + b)n = Important Formulae: Set Theory | CSAT Preparation - UPSC

Important Formulae: Set Theory | CSAT Preparation - UPSC
Some basic properties

EduRev's Tip:

  • There is one more term than the power of the exponent, n. That is, there are terms in the expansion of (a + b)n.
  • In each term, the sum of the exponents is n, the power to which the binomial is raised.
  • The exponents of a start with n, the power of the binomial, and decrease to 0. The last term has no factor of a. The first term has no factor of b, so powers of b start with 0 and increase to n.
  • The coefficients start at 1 and increase through certain values about “half”-way and then decrease through these same values back to 1.
  • To find the remainder when (x + y)n is divided by x, find the remainder when yn is divided by x.
  • (1+x)n ≅ 1 + nx, when x<<1
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FAQs on Important Formulae: Set Theory - CSAT Preparation - UPSC

1. What are the basic operations in set theory?
Ans. The basic operations in set theory include union, intersection, difference, and complement. The union of two sets A and B, denoted as A ∪ B, contains all the elements that are in either A or B. The intersection of two sets A and B, denoted as A ∩ B, contains all the elements that are common to both A and B. The difference of two sets A and B, denoted as A - B, contains all the elements that are in A but not in B. The complement of a set A, denoted as A', contains all the elements that are not in A from the universal set.
2. What is the cardinality of a set?
Ans. The cardinality of a set refers to the number of elements it contains. It represents the size or count of the set. The cardinality of a set A is denoted as |A|. For example, if set A = {1, 2, 3}, then the cardinality of A is 3, as it contains three elements.
3. What is the power set of a set?
Ans. The power set of a set is a set that contains all possible subsets of the original set, including the empty set and the set itself. For example, if set A = {1, 2}, then the power set of A, denoted as P(A), is {{}, {1}, {2}, {1, 2}}. The power set always has 2^n elements, where n is the cardinality of the original set.
4. What is the difference between a subset and a proper subset?
Ans. A subset is a set that contains all the elements of another set, including the possibility of being equal to that set. On the other hand, a proper subset is a subset that contains all the elements of another set but is not equal to that set. In other words, a proper subset is a strict subset, excluding the case where both sets are equal.
5. What is the Universal Set in set theory?
Ans. The Universal Set in set theory is the set that contains all the elements under consideration, typically denoted by the symbol U. It represents the largest possible set in a given context. All other sets discussed within the context of a problem or situation are subsets of the Universal Set. The Universal Set is often used to define the complement of a set and to establish the boundaries of set operations.
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