Sets | Mathematics for NDA PDF Download

Important Points

• Union of the sets A and B, denoted A ∪ B, is the set of all objects that are a member of A, or B, or both. The union of {1, 2, 3} and {2, 3, 4} is the set {1, 2, 3, 4}.
• Intersection of the sets A and B, denoted A ∩ B, is the set of all objects that are members of both A and B. The intersection of {1, 2, 3} and {2, 3, 4} is the set {2, 3}.
• Set difference of U and A, denoted U \ A, is the set of all members of U that are not members of A. The set difference {1,2,3} \ {2,3,4} is {1}, while, conversely, the set difference {2,3,4} \ {1,2,3} is {4}. When A is a subset of U, the set difference U \ A is also called the complement of A in U. In this case, if the choice of U is clear from the context, the notation Ac is sometimes used instead of U \ A, particularly if U is a universal set as in the study of Venn diagrams.
• Symmetric difference of sets A and B, denoted A △ B or A ⊖ B, is the set of all objects that are a member of exactly one of A and B (elements which are in one of the sets, but not in both). For instance, for the sets {1,2,3} and {2,3,4}, the symmetric difference set is {1,4}. It is the set difference of the union and the intersection, (A ∪ B) \ (A ∩ B) or(A \ B) ∪ (B \ A).
• Cartesian product of A and B, denoted A × B, is the set whose members are all possible ordered pairs (a,b) where ais a member of A and b is a member of B. The cartesian product of {1, 2} and {red, white} is {(1, red), (1, white), (2, red), (2, white)}.
• Power set of a set A is the set whose members are all possible subsets of A. For example, the power set of {1, 2} is{{}, {1}, {2}, {1,2}}.

Set Theory Formulas (Notations) 

For a group of two sets: 

• De Morgan’s laws: De Morgan’s laws are a pair of transformation rules relating the set operators “union” and “intersection” in terms of each other by means of negation.
• Demorgan’s First Law: (A ∪ B)’ = (A)’ ∩ (B)’
• The first law states that the complement of the union of two sets is the intersection of the complements
• Demorgan’s Second Law: (A ∩ B)’ = (A)’ ∪ (B)’
• The second law states that the complement of the intersection of two sets is the union of the complements.
• Cartesian product and relation:
  Cartesian product: If A and B are sets, the Cartesian product of A and B is the set
  A × B = {(a, b):(a ∈ A) and (b ∈ B)}.
• The following points are worth special attention: The Cartesian product of two sets is a set, and the elements of that set are ordered pairs. In each ordered pair, the first component is an element of A, and the second component is an element of B.
• Relation: A relation from a set A to a set B is a subset of A×B. A (binary) relation on A is a subset of A × A.
• Equivalence relation: A given binary relation ‘~’ on a set X is said to be an equivalence relation if and only if it is reflexive, symmetric and transitive.
  Equivalently, for all a, b and c in X:
  a ~ a. (Reflexivity);
  if a ~ b then b ~ a. (Symmetry);
  if a ~ b and b ~ c then a ~ c. (Transitivity)
• Representation of real numbers on a line: 
  Real numbers (R): Every number, which is either rational or irrational, is called a real number. In other words, a set of real numbers is the union of set of rational numbers and set of irrational numbers.
  Rational numbers: Any number which can be expressed in the form of a/b, where a and b both are integers and b is not equal to zero, is a rational number. Irrational numbers: The numbers which are not rational, are called irrational numbers.