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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