Question 1. What Is Discrete Mathematics?
Discrete Mathematics is a department of arithmetic regarding discrete factors that makes use of algebra and mathematics. It is increasingly more being implemented in the realistic fields of mathematics and pc science. It is a very good device for enhancing reasoning and hassle-solving competencies.
Question 2. What Are The Categories Of Mathematics?
Mathematics can be broadly categorised into classes −
Continuous Mathematics − It is primarily based upon continuous range line or the real numbers. It is characterized by means of the truth that between any two numbers, there are almost always an endless set of numbers. For example, a characteristic in non-stop arithmetic can be plotted in a smooth curve without breaks.
Discrete Mathematics − It involves wonderful values; i.E. Between any points, there are a countable wide variety of factors. For instance, if we've a finite set of objects, the function can be described as a list of ordered pairs having these objects, and may be presented as a entire list of these pairs.
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Question 3. What Is Sets In Discrete Mathematics?
A set is an unordered series of various elements. A set may be written explicitly through listing its elements using set bracket. If the order of the elements is modified or any element of a set is repeated, it does not make any adjustments within the set.
Some Example of Sets
A set of all positive integers
A set of all of the planets in the solar system
A set of all of the states in India
A set of all the lowercase letters of the alphabet
Question 4. In How Many Ways Represent A Set?
Sets may be represented in two ways −
Roster or Tabular Form: The set is represented through listing all the elements comprising it. The factors are enclosed within braces and separated by using commas.
Example 1 − Set of vowels in English alphabet, A=a,e,i,o,uA=a,e,i,o,u
Example 2 − Set of abnormal numbers much less than 10, B=1,3,5,7,nine
Set Builder Notation: The set is defined with the aid of specifying a assets that elements of the set have in not unusual. The set is defined as A=x:p(x)A=x:p(x)
Example 1 − The set a,e,i,o,ua,e,i,o,u is written as- A=x:x is a vowel in English alphabetA=x:x is a vowel in English alphabet
Example 2 − The set 1,three,five,7,nine1,three,5,7,nine is written as -B=x:1≤x<10 and (x%2)≠0
Question 5. Explain Some Important Sets?
N − the set of all natural numbers = 1,2,3,four,.....
Z − the set of all integers = .....,−three,−2,−1,0,1,2,three,.....
Z+ − the set of all wonderful integers
Q − the set of all rational numbers
R − the set of all real numbers
W − the set of all entire numbers
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Question 6. What Is Cardinality Of A Set?
Cardinality of a hard and fast S, denoted by usingSvariety of factors of the set. The quantity is likewise referred as the cardinal number. If a set has an endless wide variety of factors1,4,three,five=41,2,three,four,five,…
If there are units=two sets X and Y having same cardinality. It happens when the range of elements in X is precisely identical to the variety of elements in Y. In this example, there exists a bijective functionYless than or equal to set Y’s cardinality. It occurs whilst variety of elements in X is less than or identical to that of Y. Here, there exists an injective feature denotes that set X’s cardinality is less than set Y’s cardinality. It happens whilst number of factors in X is less than that of Y. Here, the function ‘f’ from X to Y is injective feature but notXunits X and Y are typically referred as equal sets.
Question 7. What Are The Types Of Sets?
Sets may be categorised into many types. Some of that are finite, infinite, subset, regularly occurring, proper, singleton set, etc.
Finite Set: A set which includes a particular range of factors is known as a finite set.
Infinite Set: A set which contains countless number of elements is known as an limitless set.
Subset: A set X is a subset of set Y (Written as X⊆Y) if each detail of X is an element of set Y.
Proper Subset: The term “right subset” can be defined as “subset of but not identical to”. A Set X is a proper subset of set Y (Written as X⊂YX⊂Y) if each element of X is an element of set Y and a set of all factors in a specific context or utility. All the sets in that context or application are basically subsets of this established set. Universal sets are represented as UU.
Empty Set or Null Set: An empty set carries no elements. It is denoted by means of ∅. As the variety of factors in an empty set is finite, empty set is a finite set. The cardinality of empty set or null set is zero.
Singleton Set or Unit Set: Singleton set or unit set consists of best one element. A singleton set is denoted with the aid of s.
Equal Set: If units contain the identical factors they may be stated to be equal.
Equivalent Set: If the cardinalities of sets are identical, they may be referred to as equal sets.
Overlapping Set: Two units that have at least one not unusual detail are known as overlapping units.
In case of overlapping units −
Disjoint Set: Two units A and B are known as disjoint units in the event that they do not have even one detail in not unusual. Therefore, disjoint units have the following homes −
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Question eight. What Is Set Operations?
Set Operations consist of Set Union, Set Intersection, Set Difference, Complement of Set, and Cartesian Product.
Set Union: The union of sets A and B (denoted with the aid of A∪B) is the set of elements which might be in A, in B, or in both A and B. Hence, A∪B=x∈A OR x∈B.
Set Intersection: The intersection of sets A and B (denoted by A∩B) is the set of elements which are in each A and B. Hence, A∩B=x∈A AND x∈B.
Set Difference/ Relative Complement
The set distinction of sets A and B (denoted via A–B) is the set of factors which are most effective in A however not in B. Hence, A−B=x.
Complement of a Set: The complement of a fixed A (denoted by using A′A′) is the set of elements which aren't in set A. Hence, A′=x.
More mainly, A′=(U−A) where U is a regular set which contains all objects.
Question 9. What Is Power Set?
Power set of a hard and fast S is the set of all subsets of S together with the empty set. The cardinality of a electricity set of a hard and fast S of cardinality n is 2n. Power set is denoted as P(S).
Example −For a set S=a,b,c,d allow us to calculate the subsets −
Subsets with zero elements − ∅ (the empty set)
Subsets with 1 detail − a,b,c,d
Subsets with 2 factors − a,b,a,c,a,d,b,c,b,d,c,d
Subsets with 3 elements − a,b,c,a,b,d,a,c,d,b,c,d
Subsets with four factors − a,b,c,d
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Question 10. What Is Partitioning Of A Set?
Partition of a set, say S, is a group of n disjoint subsets, say P1,P2,…Pn that satisfies the subsequent 3 conditions −
Pi does now not comprise the empty set. [Pi≠∅ for all 0<i≤n]
The union of the subsets should same the whole unique set. [P1∪P2∪?∪Pn=S]
The intersection of any wonderful units is empty.[Pa∩Pb=∅, for a≠b where n≥a,b≥0]
Question eleven. What Is Bell Numbers?
Bell numbers provide the be counted of the wide variety of methods to partition a set. They are denoted by way of Bn wherein n is the cardinality of the set.
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Question 12. What Is Discrete Mathematics Relations?
Whenever sets are being mentioned, the relationship between the elements of the units is the subsequent element that comes up. Relations may additionally exist between items of the same set or among objects of or extra sets.
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Question 13. What Is Discrete Mathematics Functions?
A Function assigns to each detail of a set, exactly one element of a associated set. Functions locate their software in various fields like illustration of the computational complexity of algorithms, counting objects, observe of sequences and strings, to call some. The third and very last chapter of this component highlights the critical aspects of functions.
Question 14. What Is Composition Of Functions?
Two capabilities f:A→Bf:A→B and g:B→Cg:B→C can be composed to give a composition gof. This is a function from A to C defined through (gof)(x)=g(f(x))
Question 15. What Is Propositional Logic?
A proposition is a group of declarative statements that has both a truth fee "genuine” or a reality price "false". A propositional consists of propositional variables and connectives. We denote the propositional variables via capital letters (A, B, and many others). The connectives join the propositional variables.
Some examples of Propositions are given underneath −
"Man is Mortal", it returns fact price “TRUE”
"12 + 9 = 3 – 2", it returns truth price “FALSE”
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Question 16. What Are Connectives?
In propositional logic normally we use 5 connectives that are −
Negation/ NOT (¬)
Implication / if-then (→)
If and best if (⇔).
OR (∨) − The OR operation of two propositions A and B (written as A∨B) is true if at least any of the propositional variable A or B is actual.
Question 17. What Are Tautologies?
A Tautology is a formulation that is continually true for each price of its propositional variables.
Example − Prove [(A→B)∧A]→B is a tautology
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Question 18. What Are Contradictions?
A Contradiction is a formula that's usually fake for each value of its propositional variables.
Example − Prove (A∨B)∧[(¬A)∧(¬B)] is a contradiction
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Question 19. What Is Contingency?
A Contingency is a formula which has both some true and some fake values for each fee of its propositional variables.
Example − Prove (A∨B)∧(¬A) a contingency
Question 20. What Are Propositional Equivalences?
Two statements X and Y are logically equal if any of the following situations preserve −
The fact tables of each declaration have the identical fact values.
The bi-conditional declaration X⇔Y is a tautology.
Example − Prove ¬(A∨B)and[(¬A)∧(¬B)] are equal
Question 21. What Is Duality Principle?
Duality principle states that for any actual declaration, the twin announcement received by means of interchanging unions into intersections (and vice versa) and interchanging Universal set into Null set (and vice versa) is likewise genuine. If dual of any statement is the declaration itself, it's miles said self-dual assertion.
Example − The dual of (A∩B)∪C is (A∪B)∩C
Question 22. What Are The Types Of Normal Forms?
We can convert any proposition in everyday paperwork −
Conjunctive Normal Form: A compound announcement is in conjunctive normal shape if it's miles obtained by means of operating AND among variables (negation of variables blanketed) linked with ORs. In terms of set operations, it's far a compound statement acquired with the aid of Intersection among variables related with Unions.
Disjunctive Normal Form: A compound assertion is in conjunctive normal shape if it is obtained by using running OR among variables (negation of variables protected) connected with ANDs. In terms of set operations, it's miles a compound assertion acquired via Union among variables related with Intersections.
Question 23. What Is Predicate Logic?
A predicate is an expression of one or extra variables described on a few unique area. A predicate with variables may be made a proposition by using either assigning a value to the variable or by using quantifying the variable.
The following are a few examples of predicates −
Let E(x, y) denote "x = y"
Let X(a, b, c) denote "a + b + c = 0"
Let M(x, y) denote "x is married to y"