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Discrete Mathematics Homework Help | Software Engineering Mathematics

Introduction And Propositions

In this tutorial, we will study the heart of discrete mathematics:

  • propositional logic: making statements

  • set theory: describing collections of objects

  • predicate logic: making statements about objects

  • relations, functions, sequences: describing relationships between objects

  • recursion and induction: reasoning about repeated application (and returning definitions

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Contents

  • Discrete mathematics

  • The Z notation

  • Propositions

  • Tautologies

  • Equivalences


The Z notation

The syntax and semantics that we choose for discrete mathematics is that of the Z notation:

  • The logic is typed: every identifier in our mathematical document is associated with a unique basic set

  • Functions are partial by default: the result of applying a function to a particular object may be undefined

  • The various sub-languages are precisely defined: a Z document is easily parsed and type-checked

Propositions

A proposition is a statement that must be either true or false Note that we deal with a two-valued logic (cf. SQL) Propositions may be combined using logical connectives The meaning of a combination is determined by the meanings of the propositions involved.


Examples

  • 2 is even

  • 2 + 2 = 5

  • tomorrow = tuesday

  • she is rich

  • he is tall

  • 2 / 0 = 0

Examples

  • ¬ (2 is an even number)

  • she is rich ∧ he is tall

  • the map is wrong ∨ you are a poor navigator

  • (2 + 2 = 5) ⇒ (unemployment < 2 million)

  • (tomorrow = tuesday) ⇔ (today = monday)


Truth Table

We use truth tables to give a precise meaning to our logical connective

Practice Example:











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