Core topics - Intro to Discrete Math
Trees: -
In graph theory, trees are the graphs which are undirected and no two vertices can be connected by more than one path. It is used to solve various complex real-life problems.
Sequence: -
Mathematically, sequence is f:N→A (a function f from natural number to a set A). In other words, sequence is a pattern between elements of a set in an ordered way. For example, 1, 3, 32, 33………
Series: -
Summation of sequence is known as series. It is denoted by the symbol sigma. Arithmetic series, geometric series, and Fibonacci series are some examples for that.
Boolean Algebra: -
Boolean algebra uses binary variables and deals with operation on logical statements, that is how a logical statement can validate itself using a truth table.
Sets: -
Sets can be defined as a “defined collection of any particular thing” which means that collection should be the same to everyone. For example a collection of top 10 writers is not an asset but a collection of books of Shakespeare is a set.
Graph theory: -
Graph theory is all about the set of points called vertices or nodes connected by lines known as paths. It is widely used to solve the complex strategic problems.
Group Theory: -
A mathematical operation with a set follows some properties, then that set is called a group under that operation. Those properties are Closure, Associative, Identity element, and existence of Inverse.
Permutation and Combination: -
This concept in math is used to find the number of ways to arrange or select certain things. For example, find the number of ways to select 3 boys out of a group of 7 boys.
Mathematical logic: -
In this, we have some set of statements to check whether they are logical, correct or not. For that we have some set of rules in order to check if that statement makes some sense or not according to those rules.
Relation and function: -
Relation and function are the subset of the cartesian product of two sets, in relation we have some common property followed by the elements and function are a special type of relation in which every element has a unique image.
Counting Theory: -
When we have to find the total number of ways in which a particular scenario can be done we have to count each and every possibility in such cases counting theory comes in handy. Permutation and combination are the two methods which help in counting those possibilities.
Sample Discrete Math Practice Problems:
Q1. Are the statements (P ∨ Q) → R and (P → R) ∨ (Q → R) logically equivalent?
Sol.
P | Q | R | (P ∨ Q) → R | (P → R) ∨ (Q → R) |
T | T | T | T | T |
T | T | F | F | F |
T | F | T | T | T |
T | F | F | F | T |
F | T | T | T | T |
F | T | F | F | T |
F | F | T | T | T |
F | F | F | T | T |
the statements are not logically equivalent.
Q2. Is it possible for a planar graph to have 6 vertices, 10 edges and 5 faces? Explain
Sol. No.
A (connected) planar graph must satisfy Euler's formula
V – E + F = 2
V – E + F = 2
Here V – E + F = 6 – 10 + 5 = 1
So, it is not possible
Discrete Math Sample Questions:
Question 1: Recurrence relation is formed of:
(a) Rules to generate the next term in the sequence
(b) Current Value
(c) Starting Value
(d) Both (a) and (b)
Answer: (c)
Explanation: The recurrence relation is made of starting value.
Question 2: Find the recursive formula for the sequence 3,6,12,24,48,96…..
Answer: (a)
Explanation: Use the recursive formula.
Question 3: An algebraic structure ____________is called a semigroup.
Answer: (a)
Explanation: The ones which satisfy closure property and associative property are called as semi-group.
Question 4: {1,-1,I,-1} is ______________under multiplication.
(a) Semi Group
(b) Cyclic Group
(c) Abelian Group
(d) All of the above
Answer: (d)
Explanation:
Question 5: Let ( Z, *) is a group with a*b = a+b-2 then inverse of a is:
Answer: (a)
Explanation:
Question 6: For semigroup homomorphism the condition should be
Answer: (d)
Explanation: Isomorphism is the condition for …….
Question 7: Let H and K be any two subgroup of a group G. H???? Ụ K is a subgroup on_________.
Answer: (b)
Explanation: H???? Ụ K is a subgroup if one is contained in other.
Question 8: In the set of integers with operation * defined by a*b = a+b-ab, the value of 3*(4*5) is:
(a) 25
(b) 15
(c) 10
(d) 5
Answer: (a)
Explanation: 3*(4*5) = 3*(4+5-20) = 3*(-11) = 3 + (-11) -3(-11).....
Question 9: If a, b are positive integers, define a*b = a where ab= a(modulo 7), with this * operation, then inverse of 3 in group G = { 1,2,3,4,5,6} is___________.
(a) 3
(b) 1
(c) 5
(d) 4
Answer: (c)
Explanation: 3 * 1 = 3
3 * 2 = 6....
Question 10: (Z*) is a group in which x*y = x+y -2, find the inverse of x:
Answer: (a)
Explanation:
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Discrete mathematics is the branch of mathematics that deals with objects that can assume only distinct, separated values such as integers. This branch of mathematics gained popularity only in the last few decades due to the growth and dependency of the world on computers and information technology. Discrete mathematics emerged as a separate area of study beginning late 1960s at the upper graduate student’s level.
The introduction to discrete mathematics in primary and secondary schools (K-12) in the United States was encouraged and facilitated by the recommendations of the National Council of Teachers of Mathematics in the curriculum and evaluation standards of school mathematics 1989).
Some of the common and everyday applications of discrete mathematics include computer run software that uses binary math. Network connections are discrete structures, online maps such as Google & Apple use discrete math to calculate the fastest routes & times. Railway planning also uses discrete math.
Discrete math was usually a course opted by the students of computer science. Therefore, it is also known as “math for computer science.” It is also related to the fields of cryptology, automated theorem proving, programming language, and software development and plays a significant role in big data analytics.
So in the 21st century, where understanding of computer codes and programming languages is a necessity for everyone, studying and understanding concepts of Discrete Mathematics is a must.
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The following topics are covered in Discrete Math:
- Set theory and notation
- Algorithms
- Computing modules, Basic number theory, and integer definitions
- Combinatorics
- Algebraic Structures
- Topology
- Discrete analysis or discrete calculus
- Game theory
- Utility Theory
- Decision Theory
- Social Choice Theory
- Operations Research
- Number Theory
- Logic
- Information Theory
- Representation of Integers
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Is discrete math difficult?
Discrete Math is not difficult but requires practice on the part of the student. If you are finding this subject difficult, you must :
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What is a bijection in discrete math?
If there is one to one correspondence between the elements of two sets under a function then we say that the function is bijective.