Loading....
Coupon Accepted Successfully!

 

Summary

  • A set is a well-defined collection of objects. Each object is called an element of the set. Usually sets are denoted by a capital letter, like ABCD.
  • In roster form of representing a set, all elements of the set are listed within curly brackets {} and separated by a comma.
  • In set-builder form, a set is described by the characterizing property of its elements.
  • The order of elements in a set is not relevant. Two identical elements are considered to be one element.
  • The set which has no elements in it is known as a null set.
  • A set containing only one element is known as a singleton set.
  • A set which has all possible elements of a situation or problem under consideration is known as a universal set.
  • A set ‘A’ will be a subset of ‘B’, if all elements of A are there in B. It is denoted by A  B. If A is a subset of B but is not equal to B, it is called a proper subset and is denoted by A  B.
  • If set A is a proper subset of B, then B will be the superset of A and is denoted by B  A.
  • The set of all subsets of a set A is called power set of A. It is denoted by P(A).
  • Two sets are said to be equal sets if each and every element in both the sets is same.
  • Two sets are said to be equivalent if they have the same number of elements.
  • When the set of elements of both A and B combined, it is said to be the union set of A and B. It is denoted by A  B.
  • The set of elements that are in A as well as in B is said to be the intersection set of A and B and it is denoted by A  B.
  • If there are no common elements in sets A and B, then the sets are said to be disjoint sets.
  • The set of elements which are in U (universal set) but not in A is said to be the complementary set of A and it is denoted by A′ or AC.
  • n(A  B) = n(A) + n(B) - n(A  B)
  • n(A  B  C) = n(A) + n(B) + n(C) - n(A  B) - n(B  C) - n(C  A) + n(A  B  C)
  • The Cartesian product of set A and set B is denoted by A × B and is defined as: A × B = {(a, b) : a  A and b  B}
  • Every subset of the product set A × B is called a relation between A and B. It is denoted by R.
  • The set of all first elements of the ordered pairs that belong to R is known as the domain of the relation R.
  • The set of all second elements of the ordered pairs that belong to R is known as the range of the relation R.
  • The relation R on a set A is said to be a reflexive relation, if R contains all possible ordered pairs of the type (xx) for all x  A.
  • The relation R on a set A is said to be symmetric, if (x, y)  R and (y, x)  R.
  • The relation R on a set A is said to be transitive if (x, y)  R and ( y, z)  R  (x, z)  R.
  • The relation R on a set A is said to be an equivalence relation, if R is reflexive, symmetric and transitive.
  • If R is a relation from set A to B, then the inverse relation R-1 from B to A is defined by R-1 = {(b, a) : (a, b)  R}
  • If each element x of A is related to a unique element f (x) of B, it is called a function from A to B and it is denoted by fA  B
  • One-one function: Each element from set A has a distinct image in set B.
  • Many-one function: Two or more than two elements of set A have the same image in B.
  • Onto function: No element in B should be without a pre-image. For onto functions, range = co-domain.
  • Into function: At least one element in B does not have a pre-image in A.




Test Your Skills Now!
Take a Quiz now
Reviewer Name