Mathematics

Sum Rule If sets A and B are disjoint sets(the sets which have no common element in them), then  |A⋃B|=|A| + |B|  For example: a class has 43 girls and 54 boys total of the class=43+54=97 Product Rule If set A and set B the cartesian product is : A*B={(a,b)| a  ∈ A and b  ∈ B} if |A| =n and |B| = m then |A*B|=n*m for example: A={a,b} and B={c,d}   A*B={(a,c),(a,d),(b,c),(b,d)}  Examples on counting: 1. how many password satisfy the following requirements: it should be 6 to 8 characters long. starts with a letter. case sensitive. other characters: digits or letters. solution: first we will find the six character password as: =52*62*62*62*62*62 { here first 52 is for 26 small letters and 26 for capital letters, other 5 times 62 are for digits(10) and letters(52=26+26)} =52*62⁵ second for seven characters password as: =52*62* 62*62*62*62*62{ here one extra 62 is multiplied according to the 1st case it is for the seventh characters} = 52*62⁶ last for eight...

 The relation is an association between two or more things. relation between sets is the association between then that defines how the sets are related.  Types of relations: Reflexive relation If a set A have elements like {a,b,c,d} the relation, must contain all (a,a), (b,b),(c,c),(d,d) in it then the relation is called as reflexive relation. in simple words the relation which contain (a,a) pair for all a n=belongs to the set, then the relation  on the set is called as reflexive relation. each element present in the set must be related to it self then the relation is called as reflexive relation. for example: A={1,2,3,6,8} the relation on set A is such that it defines (a,b) belongs to A when b is divisible by a. here R={(1,1),(1,2),(1,3),(1,6),(1,8),(2,2),(2,6),(2,8),(3,3),(3,6),(6,6),(8,8)}  here in the above relation all pairs that are like (a,a) are present in relation R that are: (1,1),(2,2),(3,3),(6,6),(8,8) all are present in the relation set so the above rela...

  directed graph A graph of the given relation. Let A and B are the two finite sets and R is the relation from A to B. For a graphical representation of a relation on a set is represented by a point. Each element in a set is represented by a point in diagraph. Those points are called as nodes. An edge is drawn from one point to another related point. This edges are called as arc. The direction in the diagraph is represented by an arrow. And all arrows in directed graph with atrows called as directed arcs. For example: Draw a  directed graph that represents the relation: R={(1,1),(2,2),(1,2),(2,3),(3,2),(3,1),(3,3)} The loops present in the diagraph represents the reflexive property that are (1,1),(2,2),(3,3) the edge from 1 to 2 represents (1,2) and similarly other edges are shown. Hasse diagram  Hasse diagram is a diagram which represents partial ordered set in the form of graph without showing its transitive relation. Steps to draw a Hasse diagram: Start with a directed...

The set in mathematics is the group of objects and the objects should be distinct. Repetition of objects is not allowed in sets.    The sets are used to define the concepts of relations and functions. the theory of sets is developed by German mathematician Georg Cantor (1845-1918). Definition of Set A set is any collection of distinct in simple way set is a collection of objects specified in such a way that we can determine whether a given object is present in the set or not.  for example : 1. A set of vegetables      2. A set of numbers 3. Set of flowers  4. A set of keys    Elements of a set  The different objects or members present in the set are called as elements of set. The elements of a set are written in any order . the elements present in the set should not be repeated. the elements are denoted by small letters.  Notation  A set is usually denoted by capital letters and the elements of the set are denoted by sma...