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Discrete mathematics is a broad range of study of mathematical structures, especially objects that have distinct and separate values. There are two forms of data; namely discrete data and continuous data. Continuous data cannot be counted but can typically be measured, while Discrete data can be large but countable too. Discrete mathematics is also otherwise known as Finite mathematics or Decision mathematics and it does not comply with continuity equations.
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Key Takeaways: Discrete mathematics, Set theory, Graph-theory, Permutation, Combination, Logic, Sequence & series, Continuity equation, Continuous data
Discrete Mathematics
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Discrete mathematics, also otherwise known as Finite mathematics or Decision mathematics, digs some of the very vital concepts of class 12, like set theory, logic, graph theory and permutation and combination. In simple words, discrete mathematics deals with values of a data set that are apparently countable and can also hold distinct values. It is also called discrete data. Some of them are formal language, types of matrices, finite graphs, trees and even integers, among many others.
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Types of Discrete Mathematics
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Discrete mathematics can be classified into several categories, like,
Set Theory
Set Theory, a broadly discussed part in mathematics, is the study of sets. Usually, set theory is represented by the braces {} symbol when defining a group of sets, or objects. In other words, an instance of a set of 3 odd numbers is {1, 3, 5, 9}. Set theory is often denoted by the primary binary operations between an object M and set A.
Graph Theory
Graph theory, being one of the fundamental subjects of discrete mathematics, is a defined study that deals with graphs. Graph is a mathematical tool that is generally used to pair a relation between two distinct objects or structures. Graphs are one of the most significant parts of discrete mathematics.
Logic
Mathematically, logic can be simply defined as the study that deals with logical reasoning. Logic is classified into three major forms of logic gates, which are AND(∧), NOT(~) and OR(∨). Each of them are widely discussed in Computer science as well. The truth values found often form a finite set.
Permutations and Combinations
Permutation can be mathematically formed when different orders can be made by a given set in a specific sequence. For instance, the set {1, 2, 3} can have several structures within the same set, such as {1, 2, 3}, {2, 1, 3}, {3, 2, 1}, {2, 3, 1}, {2, 1, 3}, and {1, 3, 2}.
Sequence and Series
When a set of numbers is particularly arranged in a definite orderly manner, it is known as sequence. While, a series forms when we find the sum of the numbers of a sequence. Series is when we add the different values given in a set.
Applications of Discrete Mathematics
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There are several applications of discrete mathematics, some of them are:
- The number theory we know today has several applications, especially in areas of cryptanalysis and cryptography.
- It is widely used in networking and determining locations of online maps, like the one Google provides.
- Discrete mathematics plays a major role in software development. It plays a pivotal role in areas of computer science.
- Discrete mathematics is also widely used in relational databases, logistics, and computing algorithms.
Use in Real World of Discrete Mathematics
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Discrete mathematics has several uses in the real world.
- It is more prevalent in numerous areas of academia, as much as industrial facets.
- Parts of encryption and decryption in areas of cryptography are highly applicable sections of discrete mathematics.
- Apart from that, computer graphics use linear algebra which is largely a vital part of discrete mathematics.
- It is also broadly used in storing electronic health care records.
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Things to Remember
Following are some important points:
- Discrete mathematics is the study of objects and structures that are discrete in nature, and not continuous.
- Discrete mathematics deals with different mathematical structures, especially objects that have distinct and separate values.
- There are many parts of discrete mathematics, like logic, set theory, graph theory, trees, permutation and combination.
- The usage of discrete mathematics is famous in several areas of computer science, much like in regions of programming languages, computer algorithms and cryptography.
Sample Questions
Ques: Evaluate the ways in which three presents can be shared among a group of four friends such that it fulfils the following conditions,
i. Nobody gets more than one present at a time.
ii. A friend can potentially get any number of presents. [2 Marks]
Ans: i. The present in the first case can be distributed in 4 different ways such that one person does not get more than one present. Now, the other two presents can be distributed respectively in 3 and 2 ways, as in,
The total number of ways the present can be distributed = 4 * 3 * 2 = 24
The answer is 24.
ii. Since there is absolutely no restriction, the presents can be distributed in 4 different ways.
Therefore, the total number of ways it can be distributed = 43 = 64.
Ques: Determine the value of set A = {1, 4, 9, 16, 25, . . . } which is in set-builder form. [3 Marks]
Ans: By understanding the pattern here, it is quite clear that the numbers are derived from squares of natural numbers. Therefore, it can be said,
12 = 1
22 = 4
32 = 9
42 = 16 and more like this.
Now, after determining the patterns, it can be said that,
A = {x : x can be considered as the square of a natural number}
Hence, it can be otherwise be stated as,
A = {x : x = n2 }
It is determined because = n ∈ N
Ques: Give a brief instance of equal sets by the following set of letters: LOYAL. [3 Marks]
Ans: Assuming that there are sets A and B.
Now, A can be considered the set of letters in ‘ALLOY’
B, at the same time, is the set of letters in ‘LOYAL’
Therefore, it can be said,
Hence,
A = {A,L,L,O,Y}
B = {L,O,Y,A,L}
Thus, the elements are similar in both the sets.
Hence, it can be evidently said that A = B.
Ques: Assume that in a G.P, the second term is considered 12, while the sixth term has been considered 192. Following the same, determine the 11th term. [3 Marks]
Ans: Assuming in a G.P there is a second term, ar = 12 . . . ( as case 1)
The sixth term is given as, ar5 = 192 . . . (as case 2)
Now, after dividing the two terms side by side, we will get,
ar5 / ar = 192/12
⇒ r4 = 16
⇒ r = 2
Now, we require to replace the values as per, r in (1), such that we get, a×2 = 12
⇒ a = 12/2 = 6
Thus, the 11the term can be given by ar10 = 6 × 210 = 6144
Therefore, the 11th term is 6144.
Ques: Determine the total sum of all four-digit numbers that can be constructed by using the number, 2, 3, 6, 9 such that no digit is repeated after used once. [2 Marks]
Ans: Three digits can be orderly arranged in 3! Ways, assuming that 2 occupies the unit’s place = 3! = 6 ways.
Hence, (3!) (2) + (3!) (20) + (3!) (200) + (3!) (2000) = 3!(2) (1111)
Now, the values that are contributed by 3, 6, 9 to the sum are as defined below,
3! (3)(1111), 3! (6) (1111), 3! (9)(1111).
Which is to say, the required sum of 3! is
3! (1111) (2 + 3 + 6 + 9)
= 1,33,320 (without any digit repeating itself in the sequence).
Ques: With the given sequence: 1, 11, 21, 31, … to 100 terms, determine the largest term common to the sequence. Along with it, also determine the following: 31, 36, 41, 46,… to 100 terms. [3 Marks]
Ans: As per the equation given, it is clear that,
The 100th term common to the sequence 1, 11, 21, 31, … can be illustrated as 1+ (100 – 1) 10 = 991
And, the 100th term common to the sequence 31, 36, 41, 46, … can be illustrated as 31 + (100 – 1) 5 = 526.
Now, it is clear that the largest common term is 526.
Hence, it can be said that, 526 = 31 + (n – 1) 10
As a result, n = 50.5
But, n = 50.
Thus, the largest common term is 31 + (n – 1) 10 = 521.
Ques: Deduce the number of every one-one function to itself from the given set, A = {1, 2, 3}. [2 Marks]
Ans: As per the given question, the set we need to evaluate is A = {1, 2, 3}.
Now, the one-one function to itself from the set A = {1, 2, 3} can simply be determined by a permutation on the three set symbols: 1, 2, and 3.
Hence, the total number of one-one maps that can be determined to itself from {1, 2, 3} will be equal to the total permutations found for three set symbols, which is 3!. The answer is 3! = 6.
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