Siddhant Volume 16, Issue 4, October-December, 2016, pp- 298-303 DOI: 10.5958/2231-0657.2016.00034.3
The Theory of Propositional Logics in Reference of Boolean Algebra Jayesh K. Tiwari1* and Rajendra Tiwari2 ABSTRACT In this article, we will prove that set of complex numbers C = a + ib, where a, b R and i = imaginary number also known as by iota (i = -1), is the set of largest number, whereas the set of natural numbers N = {1, 2, 3, …, n, …} is the set of least numbers through logical approach of validity of argument using logic of Boolean algebra and through different algebraic operations. We shall verify through rule of detachment by applying them on identity relation N W I Q R C or C R Q I W N. It will be proved by using Boolean algebra table and different Boolean algebraic properties. Keywords: Calculus, Inference calculus, Logics, Principles, Truth table, Complex numbers, Natural numbers, Boolean algebra
INTRODUCTION The rules of logic specify the meaning of mathematical statement. For instance, these rules help us reason with statement such as ‘There exists an integer that is not the sum of two squares’ and ‘ For every positive integer n, the sum of the positive integers not exceeding n is n(n + 1)/2’. Logic is the basis of all mathematical reasoning and of all automated reasoning. It has practical applications to the design of commuting machine, the specification of systems to artificial intelligence to computer programming to programming languages and to other areas of computer science as well as to many other fields. To understand mathematics, we must understand what makes up a correct mathematical argument that is a proof. Once we prove that a mathematical statement is true, we call it a theorem. A collection of theorem on topic organises what we know about that topic. To learn a mathematical topic, a person needs to actively construct
mathematical arguments on this topic and not just read exposition. Moreover, because knowing the proof of a theorem often makes it possible to modify the result to fit new situations, proofs play essential role in the development of new ideas. In fact, proofs play essential roles when we verify that computer programs produce the correct result, when we established the security of a system and when we create artificial intelligence automated reasoning system which has been constructed to allow computers to construct their own proofs. Logic is the discipline that deals with the method of reasoning. On an elementary level, logic provides rules and techniques for determining whether a given argument is valid. Logical reasoning is used in mathematics to prove theorems, in computer science to verify the correctness of programs and prove theorems in the natural and in physical sciences to draw conclusion from experiments in the social sciences and in our everyday lives to solve a multitude of problems. Indeed, we are constantly using logical reasoning.
1
Associate Professor, Department of Computer Science, Shri Vaishnav Institute of Management, Davi-Ahilya University, Indore-452009, Madhya Pradesh, India 2 Professor, Department of Mathematics, Government Madhav Science College, Vikram University, Ujjain-456010, Madhya Pradesh, India (*Corresponding author) e-mail id: *
[email protected],
[email protected] Siddhant
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The Theory of Propositional Logics in Reference of Boolean Algebra
PRELIMINARIES Now, we begin with some definitions. Definition 2.1 – Sentences: Group of words with a proper meaning is called a sentence. Definition 2.2 – Statements: All the declarative sentences to which it is possible to assign one and only one of the two possible truth values are called statements. The symbols, which are used to represent statements, are called statement letters usually the letters P, Q, R,…, p, q, r and others are used. Definition 2.3 – Logical Connectives: Logical connectives or sentence connectives are the words or symbols used to combine two sentences or statements to form a compound sentence or compound statement (Table 1). Table 1: Logical connectives with their symbols Connective Name of Symbols Word Connective Not Denial or negation And Conjunction Or Disjunction If … then Conditional Iff
Bi-conditional
Rank 1 2 3 4 5
Definition 2.4 – .Tautology: A statement formula which is true regardless of truth values of the statements which replace the variable in it is called a universally valid formula or a tautology or logical truth. Definition 2.5 – Contradiction: A statement formula which is false regardless of the truth values of the statement which replace the variable in it is called a contradiction. Definition 2.6 – Validity Using Truth Tables: Let A and B be two statement formulas. We say that ‘B logically follows from A’ or ‘B is a valid conclusion of the premise A’ iff A B, which is a tautology. Just as the definition of implication was extended to include a set of formulas rather than a single formula, we say that from a set of premises {H1, H2, H3, …, Hm} a conclusion C follows logically iff H1 H2 … Hm C. Siddhant
Given a set of premises and a conclusion, it is possible to determine whether the conclusion logically follows from the given premises by constructing truth table as follows. Let P1, P2, P3, …, Pn be the atomic variables appearing in the premises H1, H2, H3, …, Hm and the conclusion C. If all possible combinations of truth values are assigned to P1, P2, …, Pn and if the truth values of H1, H2, …, Hm and C are entered in a table, then it is easy to see from such a table whether (1) is true. We look for the rows in which all H1, H2, …, Hm have the value true (T). If, for every such row, C also has the value T, then (1) holds, otherwise fallacy exists. Definition 2.7 – Law of Syllogism: We know that the statement [(p q) (q r)] (P r) is tautology, and therefore the argument pq } premises qr …………………………..…………………………… pr (conclusion) is valid. This is called the law of syllogism. Definition 2.8 – Rule of Detachment: We know that the statement [p (p q)] q is a tautology, and therefore the argument p }
premises
pq ……......…………………………………………… q (conclusion) is a valid argument. This is called rule of detachment. This rule is also known as Modus Ponens. Algebra of Propositions The symbols used in Boolean algebra in place of symbols used in the algebra of propositions are listed in the following table:
MAIN RESULTS We take some premises; by the help of premises, we derive an argument and then we verify the validity of 299
Jayesh K. Tiwari and Rajendra Tiwari
Symbols Used in Algebra of Propositions Statements p, q, r, … F(False) T(True)
Symbols Used in Boolean Algebra + . ,(complement) Variables a, b, c, … 0 1
argument by truth table. If it is tautology, then our argument is a valid argument. We know that the structure of the real number system has evolved as a result of a process of successive extensions of the system of natural numbers. As a matter of fact, the extension becomes absolutely inevitable as the science of Mathematics developed in the process of solving problems from allied fields. As natural numbers do not consist of additive identity, so whole numbers were invented. When we add two whole numbers, we get a whole number but the inverse operation of subtraction is not always possible if we limit ourselves to the domain of whole numbers W ={0, 1, 2, 3, …} only. For instance, 7 cannot be subtracted from 3 within the system of whole numbers because there is no whole number which when added to 7 will give us 3. Means of whole numbers do not consist of additive inverse, so mathematicians discovered new numbers – that is integers. Thus, the positive and negative whole numbers together constitute the system of integers. Again we see that integers are lacking the property of multiplicative inverse. If you take two integers and multiply them, you cannot get multiplicative identity, so rational numbers came into existence. The set of rational numbers is defined by Q = {p/q; p, q I & q 0}. Clearly, rational numbers are terminating and recurring. As the basic four operations of arithmetic (division by 0 being excluded) in respect of any two rational numbers is again a rational number of course. So long, mathematicians were concerned with these four operations only. The system of rational numbers was sufficient for all purpose but the process of extracting roots of numbers (e.g. square root of 2 i.e.2, cube
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root of 7 i.e. 37 etc.) and also the desirability of giving a meaning of non-terminating and non-recurring decimal necessitated a further extension of the number system. In fact, there were lengths which could not be measured in terms of rational numbers, for instance the length of the diagonal of a square whose sides are of unit length cannot be measured in terms of rational numbers. This is equivalent to saying that ‘There is no rational number whose square is equal to 2’. So the system of rational number had been further enlarged by the introducing the so-called irrational numbers. Numbers like2, 37, , e and others are examples of irrational numbers. Rational numbers and irrational numbers together constitute the system of real numbers. The fundamental idea of extending the real number system by the introduction of complex number was first necessitated by the solution of algebraic equations. For instance, the equations x2 + 1 = 0 and x2 + 2x + 3 = 0 with real coefficients cannot be satisfied by any number in the domain of real numbers. L. Euler (1707–1783) was the first mathematician who introduced the symbol i = -1, which is called imaginary unit or Iota. A number of the form a + ib, where a, b R is called a complex number. If we write z = x + iy, then z is called a complex variable. We define a complex number as an order pair (x,y) of real numbers x and y. Thus, the complex number z = x + iy is defined as the order pair (x,y) of real numbers x and y, usually written as z = (x,y). The real numbers x and y are called real and imaginary parts of z, respectively. It is customary to write R(z) = x, I(z) = y. We shall adopt the following notations and derive following identity: N = the set of natural numbers = {1, 2, 3, …, n, …}. W = the set of whole numbers = {0, 1, 2, 3, …, n, …}. I = the set of integers = {……-3, -2, -1, 0, 1, 2, 3, …}. Q = the set of rational numbers. Ir = the set of irrational numbers. R = the set of real numbers. C = the set of complex numbers. Clearly, we observe the following identity relation: N W I Q R C or C R Q I W N.
Volume 16, Issue 4, October-December, 2016
The Theory of Propositional Logics in Reference of Boolean Algebra
Test for validity of the arguments through law of detachment.
If a set of real numbers is given, then we will derive the set of rational numbers.
If a set of complex numbers is given, then we will derive the set of real numbers.
If a set of real numbers is given, we will derive the set of rational numbers.
If a set of complex numbers is given, we will derive the set of real numbers.
Let p: if a set of real numbers is given
Let p: if a set of complex numbers is given.
We can express following argument in symbolic form as
q: we will derive the set of real numbers. We can express the following argument in symbolic form as follows: pq } premises p ………………………………………………………… q (conclusion) We shall construct the truth table for the statement and prove the statement by logical equivalence and property of Boolean algebra.
q: we will derive the set of rational numbers
pq } premises p ………………………………………………………… q (conclusion) We shall construct the truth table for the statement [(p q) p] q As we know that [(p q) p] q it s equivalent to {(p q) p} q
[(p q) p] q. As we know that [(p q) p] q, it is equivalent to {(p q) p} q
{(p q) p} q {(p q) p} q {(p. q) + p} q (p + p). ( q + p) + q ( q + p) + q ( q + q) + p 1+p=1
P
Q
q
p
(p q)
(p q) p
(p q) p q
1 1 0 0
1 0 1 0
0 1 0 1
0 0 1 1
0 1 0 0
0 1 1 1
1 1 1 1
As the last column contains only 1s, the given argument is valid.
Siddhant
{(p q) p} q {(p q) p} q {(p. q) + p} q (p + p). ( q + p) + q ( q + p) + q ( q + q) + p 1+p=1
P
q
q
p
(p q)
(p q) p
(p q) p q
1 1 0 0
1 0 1 0
0 1 0 1
0 0 1 1
0 1 0 0
0 1 1 1
1 1 1 1
As last column contains only 1s, the given argument is valid. If the set of rational numbers are given, then we will derive the set of integers. If the set of rational numbers are given, we will derive set of integers. 301
Jayesh K. Tiwari and Rajendra Tiwari
Let p: if a set of rational numbers is given
pq }
q: we will derive the set of integers We can express following argument in symbolic form as pq }
p ………………………………………………………… q (conclusion) We shall construct the truth table for the statement
premises
p ………………………………………………………… q (conclusion)
[(p q) p] q As we know that [(p q) p] q it s equivalent to {(p q) p} q
We shall construct the truth table for the statement
[(p q) p] q As we know that [(p q) p] q it s equivalent to {(p q) p} q
premises
{(p q) p} q {(p q) p} q {(p. q) + p} q (p + p). ( q + p) + q ( q + p) + q ( q + q) + p 1+p=1
P
Q
q
p
(p q)
(p q) p
(p q) p q
1 1 0
1 0 1
0 1 0
0 0 1
0 1 0
0 1 1
1 1 1
0
0
1
1
0
1
1
{(p q) p} q {(p q) p} q {(p. q) + p} q (p + p). ( q + p) + q ( q + p) + q ( q + q) + p 1+p=1
P
Q
q
p
(p q)
(p q) p
(p q) p q
1 1 0
1 0 1
0 1 0
0 0 1
0 1 0
0 1 1
1 1 1
0
0
1
1
0
1
1
As the last column contains only 1s, the given argument is valid. If a set of whole numbers is given, then we will derive the set of natural numbers.
As the last column contains only 1s, the given argument is valid.
If a set of whole numbers is given, we will derive the set of natural numbers.
If a set of integers is given, then we will derive the set of whole numbers
Let p: if a set of whole numbers is given.
If a set of integers are given, we will derive the set of whole numbers.
We can express following argument in symbolic form as pq } premises p ………………………………………………………… q (conclusion)
Let p: if a set of integers is given. q: we will derive the set of whole numbers. We can express following argument in symbolic form as 302
q: we will derive the set of natural numbers.
Volume 16, Issue 4, October-December, 2016
The Theory of Propositional Logics in Reference of Boolean Algebra
We shall construct the truth table for the statement
P
Q
q
p
(p q)
(p q) p
(p q) p q
1 1 0
1 0 1
0 1 0
0 0 1
0 1 0
0 1 1
1 1 1
0
0
1
1
0
1
1
[(p q) p] q As we know that [(p q) p] q it s equivalent to {(p q) p} q
{(p q) p} q {(p q) p} q {(p. q) + p} q (p + p). ( q + p) + q ( q + p) + q ( q + q) + p 1+p=1
As last column contains only 1s, hence the given argument is valid. Henceforth, complex number is the set of largest number and natural number is the set of least number.
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