[Assigment][Saira Kanwal]

[Discrete Mathematics]

[Mam Werda]

[19-8-2016]

Question # 1

What is Tree?

Answer

Tree

Tree is a special type of graph which is particularly important in both theory and application. Many powerful algorithms in computer science and software engineering are tree based algorithms.

Trees, Rooted Trees and Binary Treeso A nontrivial circuit is a circuit with at least one edge. A

nontrivial circuit is also called a cycle.o A tree is a connected graph without nontrivial circuits.o A forest is composed of one tree or some disconnected trees.o A terminating vertex (or a leaf) in a tree is a vertex of degree

1.o An internal vertex (or a branch vertex) in a tree is a vertex of

degree greater than 1.o Vertices are sometimes referred to as nodes, particularly when

dealing with graph trees.

Examples

Graphs

are both trees.

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Graph

is not a tree.

Question # 2

What is Venn Diagram ?

Answer

Venn Diagram

John Venn was born August 4, 1834 in Hull Yorkshire, England and died April 4 1923 in Cambridge, England.

Venn diagram (a diagrammatic way of representing sets and their relationships) were introduced in 1880 by John Venn.

Venn diagram is very useful visually the basic concept of set and relationship between set. In a Venn diagram a rectangular region represent the universal set and the region bounded by simple close curved represent other set which are a sub set of universal set.

Question # 3

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What is a Graph ?

Answer

Graph

Two-dimensional drawing showing a relationship (usually between two set of numbers) by means of a line, curve, a series of bars, or other symbols. Typically, an independent variable is represented on perpendicular axis intersect at a point called origin, and are calibrated in the units of the quantities represented. Though a graph usually has four quadrants representing the positive and negative values of the variables, usually the horizontal line (X-axis) and an dependent variable on the vertical line (Y-axis). The only the north-east quadrant is shown when the negative values do not exist or are of no interest.

Question # 4

What is pigeonhole system ?

Answer

Pigeonhole PrincipleSuppose that a flock of 20 pigeons flies into a set of 19 pigeonholes to roost. Because there are20 pigeons but only 19 pigeonholes, a least one of these 19 pigeonholes must have at least twopigeons in it. To see why this is true, note that if each pigeonhole had at most one pigeon in it,at most 19 pigeons, one per hole, could be accommodated. This illustrates a general principle called the pigeonhole principle, which states that if there are more pigeons than pigeonholes,then there must be at least one pigeonhole with at least two pigeons in it.

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If "n" number of pigeons or objects are to placed in "k" number of pigeonholes or boxes; where k<n, then there must be at least one

pigeonhole or box which has more than one object.

Let us suppose that |A||A| and |B||B| represent the total number of members in two finite sets AA and BB respectively, then there will be one-one correspondence, such that:f:Af:A ⇒⇒ BB ⇔⇔ |A|=|B||A|=|B|

Question # 5

What is function and its types ?

Answer

FunctionA function or mapping (Defined as f: X→Y) is a relationship from elements of one set X to elements of another set Y (X and Y are non-empty sets). X is called Domain and Y is called Codomain of function ‘f’. Function ‘f’ is a relation on X and Y s.t for each x ∈ X, there exists a unique y ∈ Y such that

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(x,y) ∈ R. x is called pre-image and y is called image of function f. A function can be one to one, many to one (not one to many). A function f: A→B is said to be invertible if there exists a function g: B→A

Types Of Function

One – one function (it is also called as injection function): - Function f: K → L is called as one – one function if every input term of element 'K' has different image in 'L'. So it can be written as: f : K → L is one – one if value of 'k' not equals to 'l'. (k ᚌl) → f (l) ᚌf (l) for all kl Ԑ K.

Many one function: - A function f: K → L is called as many one function if two or more elements of set 'K' have same images in 'L'. In mathematical form it can be written as:f: K → L is a many one function if there exist a, b Ԑ K such that a ᚌ b but f (a) = f (b).

Onto function: - Function f: K → L is called as Onto function or it is also called as 'surjection' if every value of element 'L' is image of some element of 'K' i.e. if f (K) = L, and range of 'f' is co – domain of function 'f' or in other words we can say that elements of 'L' has no pre – image in element 'K'. In mathematical form it can be written as:

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This is all about types of functions and its graphs.

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