### 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

Â¬ (2 is an even number)

she is rich âˆ§ he is tall

the map is wrong âˆ¨ you are a poor navigator

(2 + 2 = 5) â‡’ (unemployment < 2 million)

(tomorrow = tuesday) â‡” (today = monday)

Truth Table

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

Practice Example:

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