The Nature of Mathematics
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Mathematical Language
A language is a learned system of communication using arbitrary meanings in a
community
Universal language
Words (operations, numbers)
Letter conventions (letters used to represent particular values)
Nouns (numbers, expressions)
Verbs (=, <, >)
Pronouns (x, y, z, expressions)
Sentences (equations, true/false - DECLARATIVE - sentences)
5 + 3 = 8 (true)
-4 + 10 = 14 (false)
The truth of a declarative sentence depends on the value of a variable (x, y,
z)
x + 3 = 7 is correct only when x = 4
3x + 5x = 8x is always true
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x = x + 1 is always false
Propositional Logic/Calculus
Proposition: Declarative sentence that is either true or false but NOT both (NOT
sometimes yes/no), basic building blocks of logic
Letters can represent propositional (true/false) variables
p, q, r, s…
Truth value - Either T/F
Greek Aristotle was the first to develop it 2300 yrs ago
Compound propositions - New propositions formed from combining old ones with
logical operators
Negation: Opposite (not p)
Connectives
2^n for the number of possible combinations (n = no. of variables)
Conjunction: Combination (p /\ q)
and/but are interchangeable
only T when both are true
Disjunction: Either - or - (p v q)
F when both are false, otherwise T
T when at least one variable is true
”Find the disjunction” - The teacher was absent OR the students
took the exam (just translate into a sentence with OR)
Inclusive Or - Students who have taken calculus or computer
science can take this class
Exclusive Or - Students who have taken calculus or computer
science, BUT NOT BOTH, can enroll this class
Implication: If p then q (p → q)
p implies q
if p, q
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Only false when p is true and q is false (T & F - “promise broken”)
“If I am elected, then I will end labor contractualization.”
TT - Elected, ends labor c.
TF - Elected, doesn’t end labor c. (promise broken)
FT - Not-elected, ends labor c.
FF - Not elected, doesn’t end labor c.
Converse, contrapositive, and inverse
Converse: Interchanging the propositions (q → p)
Contrapositive: Interchanging and changing if it’s negated or not (~q
→ ~p)
Inverse: Negate both sides (~p → ~q)
Biconditional (bi-implications): p if and only if q (↔)
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True when p and q have the same truth values
Precedence of Logical Operators
1. Negation (~)
2. Combination (/\)
3. Disjunction (v)
4. Implication (→)
5. Biconditional(↔)
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IN LOGIC, GRAMMAR IS NOT A CONCERN, REPETITION IS FINE
DONT CHANGE THE TENSE
Truth Table Trick
1st column - Half T, half F
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2nd column - Alternating 2 T’s, 2F’s
3rd column - Alternating T, F
@May 17, 2023
Finding the Truth Value of Propositions
1. Substitute variables with their truth value (T/F)
2. Stop only when there are no operators
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