Discrete Mathematics Answers

Using the method of "searching for a falsifying truth assignment" prove the

De Morgan's law

If A= (a,b,c) then generate P(A) and 3 partition sets for A.

1. Construct a proof for the five color theorem for every planar graph.

2. Discuss how efficiently Graph Theory can be used in a route planning project for a vacation trip from Colombo to Trincomalee by considering most of the practical situations (such as mileage of the vehicle, etc.) as much as you can. Essentially consider the two fold,

- Routes with the shortest distance (Quick route travelling by own vehicle)

- Route with the lowest cost

1. State the Dijkstra’s algorithm for a directed weighted graph with all non-negative edge weights.

2. Function converts Fahrenheit temperatures into Celsius. What is the function for opposite conversion?

Discuss two examples on binary trees both quantitatively and qualitatively.

1. Discuss two real world binary problems in two different fields using applications of Boolean Algebra

1. Develop truth tables and its corresponding Boolean equation for the following scenarios.

i. ''If the driver is present AND the driver has NOT buckled up AND the ignition switch is on, then the warning light should turn ON.''

ii. If it rains and you don't open your umbrella, then you will get wet.

1. Describe the characteristics of different binary operations that are performed on the same set.

2. Justify whether the given operations on relevant sets are binary operations or not.

i. Multiplication and Division on se of Natural numbers

ii. Subtraction and Addition on Set of Natural numbers

iii. Exponential operation: on Set of Natural numbers and set of Integers

Part 2

1. Write the multisets of prime factors for the given numbers.

I. 160

II. 120

III. 250

2. Write the multiplicities of each element of multisets in part 2(1-I, ii,iii) separately.

3. Find the cardinalities of each multiset in part 2-1.