Sets in Mathematics JEE Mains | CBSE | Class 11

Selfstudy

Introduction – In mathematics, a set is a group of things or people of distinct objects, by looking attentively at as an object in its own right. For example, the numbers 2, 4, and 6 are different objects when considered separately, but when they are considered as a group or as a whole they form a single set of size three, which can be written as {2,4,6}. The concept of a set is one of the most forming a necessary base or core in mathematics. Developed at the end of the 19th century, set theory is now a present everywhere part of mathematics which can be used as a foundation from which nearly all of mathematics can be derived. In mathematics education, elementary topics such as Venn diagrams are taught at a young age, while more advanced concepts are taught as part of a university degree.

Mathematics-Set-Part-1-IIT-JEE-Mains-&-Advanced(1)
Mathematics-Set-Part-1-IIT-JEE-Mains-&-Advanced(1)

Type of Sets –

  • Empty set – The Empty Set or Null Set is a unique set having empty or no elements. Its size or cardinality (count of elements in a set) is zero. Some unquestionable set theories ensure that the empty set exists by including a general truth of empty set; in other theories, its state of living or having objective reality course can be traced. Many possible properties of sets are vacuously true for the empty set
  • Singleton set -A set that has only one member in it is called as a Singleton Set. In other words, it has a single element (neither less nor more). It can be represented in the following manner : S = {a}. In which S is a singleton set having an element a. A Set S can also be termed as a singleton set of a. We can also prepare to create or prepare methodically. A desire of singleton set under : S = {x : x = a}. The singleton set is also known as a unit set.
  • Finite and Infinite set – A set that has fixed number of elements or a finite number of elements are called Finite set. Like {11 ,12, 13, 14, 15, 16} is finite set whose cardinality is 16, since it has 16 elements. Otherwise, it is known as an infinite set. It can be countable or Uncountable. The union of some infinite sets are infinite and the power set of an infinite set is infinite. Examples: Set of all the weeks in a month is a finite set. Set of all integers is an infinite set.
  • Union of sets – The union of two sets A and B is the set of elements, which are in A or in B or in both. It is denoted by A ∪ B and is read ‘A union B’
  • Intersection of sets – A set of elements that are common in both of the sets. Intersection is similar to making a group of common elements. The symbol should be symbolized as ‘∩’. If X and Y are known to be two sets, then the intersection is represented as X ∩ Y and would be called as X intersection Y and mathematically, we can write it as
    X∩Y={x:x∈X∧x∈Y}Example: X = {1,2,3,4,5}
    Y = {2,3,7}
    X ∩ Y = {2,3}
  • Difference of sets – The difference of set B from set A, denoted by A-B, is the set of all the elements of set A that are not in set B. So here we are proving an example of it which can help you. Here is an example A-B = { x: x∈A and x∉B} .
  • Subset of a set – In set theory, a set X is a subset of a Set named as Y, if the set named as X is contained in set Y. Then, it means, that all the elements of the set X also belong to the set Y. It is denoted by ‘⊆’ or X ⊆ Y. Example:X = {1,2,3,4,5}
    Y = {1,2,3,4,5,7,8}
    Here, X is said to be the subset of Y.
  • Disjoint sets – If two sets having name X and Y. They should not have common elements or if the intersection of any 2 sets X and Y is the empty set, then these kinds of sets are said to be disjoint sets i.e. X ∩ Y = ϕ. This means, when this condition n (X ∩ Y) = 0 is true, then the sets are called as Disjoint Sets.
    Example:
    X = {1,2,3}
    Y = {4,5}
    n (A ∩ B) = 0.
    Therefore, these sets X and Y are disjoint sets.
  • Equality of two Sets – The two sets are called equal or same to each other if they contain the same elements. When the sets X and Y is said to be equal, if X ⊆ Y and Y ⊆ X, then we will write as X = Y.Examples:
    If X = {1,2,3} and Y = {1,2,3}, then X = Y.
    Let P = {a, e, i, o, u} and Q = {a, e, i, o, u, v}, then P ≠ Q, since set Q has element v as the additional element.

Type of Intervals –

  • Combination of SetsThe number of k-combinations for all k is the number of subsets of a set of n elements. … These combinations (subsets) are enumerated by the 1 digits of the set of base 2 numbers counting from 0 to 2n − 1, where each digit position is an item from the set of n. This is known as Combination of Set.
  • Complement of a SetWhen all sets under consideration are considered to be subsets of a given set U, the absolute complement of A is the set of elements in U but not in A. The relative complement of A with respect to a set B, also termed the difference of sets A and B, written B ∖ A, is the set of elements in B but not in A. Then it is called as Complement of Set.
Mathematics-Set-Part-1-IIT-JEE-Mains-&-Advanced
Mathematics-Set-Part-1-IIT-JEE-Mains-&-Advanced

Relations –

  • Cartesian Product -A Cartesian product is called to be known as a mathematical operation that returns or gives from multiple sets. … If the Cartesian product rows named as X columns is taken, the cells of the table contain the pair of form in order which is in the way of row value, column value.
  • Relation – A binary relation R(A, A) is a relation between A and A.This type of relation may always be composed with itself, and its inverse is also a binary relation on A. The identity relation on A is defined by an iAa for all a∈A.
  • Domain And Range of RelationThe domain is defined as all the possible input values (usually x) which allow the formula to work.  Note that values that cause a denominator to be zero, which makes the function undefined, are not allowable values.
    • The function y = 4x2 – 9 has a domain of all real numbers, which can be expressed using the interval.  Every possible x-value will give you a legitimate y-value.
    • The function has a domain of all real numbers except -5, because when x = -5, the denominator will be zero, and the function will be undefined.  We can express this using the interval.

    The range is the set of all possible output values (usually y), which result from using the formula. If you graph the function y = x2 – 2x – 1, you’ll see that the y-values begin at -2 and increase forever.  The range of this function is all real numbers from -2 onward.  We can express this using the interval.

  • Inverse Relation – The inverse relation of a binary relation is the relation that occurs when the order of the elements is switched in the relation. For example, the inverse of the relation ‘child of’ is the relation ‘parent of’. In formal terms, if X and Y are sets and LX × Y is a relation from X to Y, then L−1 is the relation defined so that y L−1 x if and only if x L y. In set-builder notation, L−1 = {(y, x) ∈ Y × X | (x, y) ∈ L}.
  • Identity Relation – Let X be a set. The Identity Relation I on Z is defined to be the relation where for all x,y∈X we have that xIy if and only if x=y. For example, consider the set of integers Z∖{0}. Define R to be the relation such that for x,y∈Z we have that xRy if xy=1. If xRy then xy=1 so x=y. Conversely, if x,y∈Z∖{0} and x=y then xy=1 so xRy. Therefore R is the identity relation on Z∖{0} so R=I
  • Equivalence Relation -An equivalence relation on a set X is a subset of X×X, i.e., a collection R of ordered pairs of elements of X, satisfying certain properties. Write “xRy” to mean (x,y) is an element of R, and we say “x is related to y,” then the properties are1. Reflexive: aRa for all a in X,2. Symmetric: aRb implies bRa for all a,b in X.  Transitive: aRb and bRc imply aRc for all a,b,c in X, where these three properties are completely independent. Other notations are often used to indicate a relation, e.g., a=b or a∼b.

 

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