# 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
*A*,*B*,*C*,*D*. - 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€˜
*A*â€™ will be a*subset*of â€˜*B*â€™, 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*A*â€² or*A*.^{C} *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*A*and b*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 (*x*,*x*) for all*x*âˆˆ*A*. - The relation
*R*on a set*A*is said to be*symmetric*, if (x,*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*f*:*A*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*.