Discrete Structure and Optimization

Function: - It is a mapping in which every element of set A is uniquely associated at the element with set B. The set of A is called Domain of a function and set of B is called co-domain. Domain: - The set of A is called Domain of function Co-Domain: - The set of B is called Co-Domain. Range: - It is a collection of the second values in the ordered pair. Types of Functions / Properties: - One-to-One (Injective): - One element of the Domain set is connected to one element of the Co-Domain set. Onto (Surjective): - Every element of the Co-domain set have one pre-image. Bijective (1:1 and onto): - Into: - If there exists even a single element in B having no pre-image in a, then f is said to be an into function.  Clearly, f is a function from A to B. Now, there exists an element C <- B, having no pre-image in A. so, f is an into function. One-one into: - f x -> y. The function f is called one-one into a function if different elements of x have different unique images ...

Ordered Pair : - An ordered pair is a pair of entries in the specified order. Domain: - The domain of the relation r is the set of all first components of the ordered pairs. Range : - It is a collection of the second values in the ordered pair. Relation: - The relation between the two sets is defined as the collection of the ordered pair, in which the ordered pair is formed by the object from each set. Representation of Relation: - Roster form : -  Set Builder form: - Equivalence relation Property: - In which satisfy these properties: - Reflexive relation Symmetric relation Transitive relation Types of Relations / Properties of Relations: - Reflexive Relation: - If every element of set A maps for itself, then set A is known as a reflexive relation. Symmetric Relation: - A relation R on a set is said to be symmetric iff Transitive relation: - A relation R on a set A is said to be transitive iff Irreflexive Relation: - Antisymmetric Relation: - Asymmetric Relation: - Identity r...

A set is an unordered collection of different elements. Sets are represented as a collection of well-defined objects or elements and it does not change from person to person. N – Set of all-natural numbers Z – Set of an integer number Q – Set of all rational number R – Set of all real number Z+ - Positive integers W – Whole number Representation of Sets: - Roster or tabular form: - In this form of representation we list all the elements of the set within braces {} and separate them by commas. Example : - the set of natural numbers less than 5 Set Builder form: - In this form of representation we list the properties fulfilled by all the elements of the set. Example : - A = {2, 4, 6, 8} Types of Set: - Empty Set: - A set that does not contain any element is called an empty set or void set or NULL set. It is denoted by {} or Ø. A set of apples in the basket of groups is an example of an empty set because in a group's basket there are no apples present. Singleton set : - A...