Discrete Mathematics: Overview, Questions, Preparation

Mathematics is fundamentally divided into two branches—continuous mathematics and discrete mathematics. Continuous mathematics is the study of real numbers whose result is uncountably infinite. Discrete mathematics, as the name suggests, is the study of distinct values whose result is either finite or countably finite. The study of discrete mathematics is very advantageous as it increases your reasoning and logical powers. Moreover, it is a high scoring part of the syllabus.

The topics that are included in discrete mathematics are:

• Set Theory
• Permutation and Combination
• Graph Theory
• Logic
• Sequence and Series

Set Theory

Sets are nothing but a collection of objects. For example, the collection of the first five natural numbers represents a set {1, 2, 3, 4, 5 }.
In this chapter, you will study concepts and definitions regarding sets. Moreover, you will get to know how to represent a set, types of sets, subsets, Venn diagrams, laws of the algebra of sets.

Read more about Set Theory

Permutation and Combination

Permutation and combination are the various ways in which objects can be selected to form sets.

We can find permutation by:

nPr = (n!) / (n-r)!

We can find combination by:
nCr = n! / r! (n-r)!

This chapter carries 5-8 marks in your class 11^th exams.

Logic

Logic is what we know as valid reasoning. Computer science also works on logic. We use logic gates to solve the problems which are AND, OR and NOT.
This carries 2 marks in class 11^th.
Graph Theory

A graph is a mathematical structure used to display a relationship between objects. It is made up of nodes that are connected by edges. Graphs make it easy to study and analyse a large amount of data.

Sequence and Series

A sequence is the arrangement of numbers in a particular order while a series is a sum of numbers arranged in order.

This carries 5-8 marks in Class 11^th. Almost every field of mathematics uses or refers to sets. It helps in solving problems based on sequences, probability and geometry. 

1.If S and T are two sets such that S has 21 elements, T has 32 elements, and S ∩ T has 11 elements, how many elements does S ∪ T have?
Solution. Given,

n(S) = 21

n(T) = 32

n(S∩T) = 11

n( S ∪ T) = n(S) + n(T) – n(S∩T)

= 21+ 32+ 11

= 42

The set ( S ∪ T) has 42 elements.

2. How many three-digit numbers can be formed using digits 1 to 9 if no digit is repeated?

Solution. Digits available= 9

Places available = 3

 ⁿP_r = (n!) / (n-r)!

 permutation = 9! / (9-3)!

=9*8*7*6! / 6!

=504

504 numbers can be formed.

3. A man starts repaying the loan with the first instalment of Rs. 100. If he increases the instalment by Rs. 5 every month, what amount will he pay in the 30^th instalment?

This forms an A.P. = 100, 105, 110, …..

first term is, a = 100

the common difference d = 5

The 30^th term of this A.P. will be,

A₃₀  = a + (30 – 1)d

= 100+ (29) (5)

= 245.

Hence, the amount the man pays in the 30^th instalment will be Rs. 245.

Q: What are the areas in which discrete mathematics can be applied?

Q: Give an example of discrete mathematics in the real world.

Q: State the formula of the sum of first n natural numbers.

Q: What are logic gates?

Q; What is the power of a set, say X?