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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 â€˜Aâ€™ will be a subset of â€˜Bâ€™, if all elements of A are there in B. It is denoted by A âŠ† 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 âŠ‚ B.
• If set A is a proper subset of B, then B will be the superset of A and is denoted by B âŠƒ 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 âˆª 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 âˆ© 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 Aâ€² or AC.
• n(A âˆª B) = n(A) + n(B) - n(A âˆ© B)
• n(A âˆª B âˆª C) = n(A) + n(B) + n(C) - n(A âˆ© B) - n(B âˆ© C) - n(C âˆ© A) + n(A âˆ© B âˆ© C)
• The Cartesian product of set A and set B is denoted by A Ã— B and is defined as: A Ã— B = {(a, b) : a âˆˆ A and b âˆˆ B}
• Every subset of the product set A Ã— 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 âˆˆ A.
• The relation R on a set A is said to be symmetric, if (x, y) âˆˆ R and (y, x) âˆˆ R.
• The relation R on a set A is said to be transitive if (x, y) âˆˆ R and ( y, z) âˆˆ R â‡’ (x, z) âˆˆ 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) âˆˆ 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 â†’ 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.