Eric Chapdelaine
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MATH1365 Introduction To Mathematical Reasoning


Introduction and Fundamentals

Definitions, Theorems, and Proofs

Mathematical Reasoning: Rigorous math based on logic

Statement: Declarative sentence that has a definite truth value

3 Types of Statements

  1. Definition: Gives precise meaning. Always true.
  2. Theorem: Have to prove to be true (8 is an even number).
  3. Proof: Logical arguments to show a theorem is true

Proposition (more modest than theorem) = result. Another word is fact.

Undefined terms/Axioms: Don’t need to explicit state in proof.

  1. Numbers: Integer, Real Number (We will define even, odd, rational, etc.)
  2. Operations: Addition, Subtraction, Multiplication, and Division. (We will prove even + even = even)
  3. Basic Properties of Arithmetic
  4. Ordering of Real Numbers

When in doubt, prove it (or ask).

Logical Connectives

NOTE:

The rows of a truth table can be determined by $2^{\text{number of compnent statements}}$

Not

Takes one component statement

Notation: $\sim p$

Truth Table:

$p$ $\sim p$
T F
F T

And

Conjunction

Takes 2 component statements

Notation: $p \wedge q$

Truth Table:

$p$ $q$ $p \wedge q$
T T T
T F F
F T F
F F F

Or

Disjunction

Takes 2 component statements

Notation: $p \vee q$

Truth Table:

$p$ $q$ $p \vee q$
T T T
T F T
F T T
F F F

If then

Conditional statement

Another way to say it is $p$ implies $q$.

Truth Table:

$p$ $q$ $p \rightarrow q$
T T T
T F F
F T T
F F T

Warning:

It doesn’t say anything about the condition when $\sim p$ is true.

$p \rightarrow q$

Note, the inverse and the converse have the same truth table and therefore they are the same.

if and only if

Also called iff or biconditional

Notation: $p \leftrightarrow q$

Truth Table:

$p$ $q$ $p \leftrightarrow q$
T T T
T F F
F T F
F F T

NOTE:

All definitions are bidirectional. That is, if a definition has “if … then”, it means “if and only if”.