Mathematical Foundation of Computer Science

DHARM 84 MATHEMATICAL FOUNDATION OF COMPUTER SCIENCE of closure property by operation + over K (since addition of two positive o ...

DHARM ALGEBRAIC STRUCTURES 85 p 1 = 123 231 132 213 312 (^23123) F H I K = F H I K = F H I ,,ppK and p 4 = 321 132 213 321 231 ( ...

DHARM 86 MATHEMATICAL FOUNDATION OF COMPUTER SCIENCE ×1ωω^2 11 ωω^2 ωωω^21 ω^2 ω^21 ω Fig 4.5 Operation Table. 4.8 Subgroups.... ...

DHARM ALGEBRAIC STRUCTURES 87 So, ê a–1 ∈ Y Y–1 ⊂ Y, thus a–1 ∈ Y. Similarly, if a, b ∈ Y ⇒ b–1 ∈ Y and ab–1 ∈ Y; So, (ab–1)–1 ∈ ...

DHARM 88 MATHEMATICAL FOUNDATION OF COMPUTER SCIENCE l Consider another example of group (X, ×), i.e. X = {1, ω, ω^2 }, where ω ...

DHARM ALGEBRAIC STRUCTURES 89 Left cosets and right cosets may or may not be equal. They are equal only for commuta- tive groups ...

DHARM 90 MATHEMATICAL FOUNDATION OF COMPUTER SCIENCE Hence, from the theorem we obtain that, if X is finite group and g ∈ X then ...

DHARM ALGEBRAIC STRUCTURES 91 Now we discuss the properties of a group homomorphism, that are, (i) Preservance of identities (ii ...

DHARM 92 MATHEMATICAL FOUNDATION OF COMPUTER SCIENCE Reader must note that if we change the first group as (R, +), then R and R+ ...

DHARM ALGEBRAIC STRUCTURES 93 Now check whether (X, •) is a monoid or not. If (X, •) is a monoid then it should have an identity ...

DHARM 94 MATHEMATICAL FOUNDATION OF COMPUTER SCIENCE l There exists precisely one element – x, i.e. x + (– x) = 0 for all x ∈ X, ...

DHARM ALGEBRAIC STRUCTURES 95 For example, ring (I, +, •) is an integral domain but ring (2I, +, •) is not an integral domain be ...

DHARM 96 MATHEMATICAL FOUNDATION OF COMPUTER SCIENCE (iv) If ring X contains more than two elements then there exist distinct el ...

DHARM ALGEBRAIC STRUCTURES 97 Form the operation table our observations is follows, l It has an identity element 1, so it is a r ...

DHARM 98 MATHEMATICAL FOUNDATION OF COMPUTER SCIENCE Example 4.19. Test the following statements are true or false. If x is an ...

DHARM ALGEBRAIC STRUCTURES 99 EXERCISES 4.1 Let X = {0, 1, 2, 3, 4} then show that, (i) Algebraic structure (X, + 5 ) is an Abel ...

THIS PAGE IS BLANK ...

PROPOSITIONAL LOGIC 5.1 Introduction to Logic................................................................................... ...

5.1 INTRODUCTION TO LOGIC Study of logic is greatly concerned for the verification of reasoning. From a given set of state- ment ...

DHARM PROPOSITIONAL LOGIC 103 manuscripted in symbolic form (object language). Arguments are prepared in natural language (Engli ...