Chapter 1: Mathematical Logic and Reasoning
Day
1
2
3
4
Subject Matter of the Day
 Lesson 1.1 Statements and Quantifiers
o Statement: a sentence either True or False not Both
o Universal Statement: Holds for all elements in some set
o Quantifier: Limits a statement
o  = For all
o Law of Substitutions: Place an element into a universal
o If p(x), then q(x) statements that contain a variable
o Counterexample: Shows a false of a statement
o Existential Statements: there exists at least one time a
statement is true
o  = There is or There exists
 Problems
Page Examples
1 26
11
1,3,4,11,16
 Lesson 1.2 Negations
o Negation: (~p)(not p) Expresses the opposite of a
statement (can be made by adding ‘It is not the case’)
o Truth Table: chart that denotes the results of a statement
o Negation of a universal is an existential statement
o Negation of an existential is an universal
o Complex: page 17
also ~(~p)= p
 Problems
Page Examples
1 25
17
1,4-6,14
 Lesson 1.3 AND and OR and De Morgan’s Laws
o Inequalities: (Review) #<x<# AND; x># x<# OR
o p and q: only true if p and q are true
o p or q: is true if p is true, q is true, or both are true
o Inclusive OR: possibility both are true
o Exclusive OR: both can not be true
o Truth Tables:
p
q
p and q
p or q
T
T
T
T
T
F
F
T
F
T
F
T
F
F
F
F
o Logical Expression: Statements combined in a clear way
with and, or, not, if-then
o Logically Equivalent: () have same truth tables
o De Morgan’s Law:
1. ~(p and q)  (~p) or (~q)
2. ~(p or q)  (~p) and (~q)
 Problems
Page Examples
1 24
24
1,10,12,13
 Ms 1.1 to 1.3
 Lesson 1.4 Computer Logic Networks
o Logic Gate: input and output wires attached to a symbol
 T  on  1
 F  off  0
o Input-Output Tables: Same as Truth Tables instead use
1&0 and output instead of the logical expression
5
6
7
8
NOT
AND
OR
o Network: Combinations of gates
o Functionally Equivalent: () Same as logically equivalent
o Boolean Algebra: any operation of two value system if
combined using (and, or, not)
 Problems
Page Examples
1 15
32
4,7
 Lesson 1.5 IF-THEN Statements
o If-Then: one statement is suppose to follow another, basis
for Deduction and Proofs
o If p then q
reads or means that p implies q
pq
hypothesis, antecedent  conclusion, consequent
o FALSE when p is True and q is False, otherwise  True
o Negation of a Conditional: ~  p  q  p and  ~ q
 NOTE (pg 37) Caution: ~conditional is an and statement
o Negations of a Universal Conditional:
 x in S, if p(x) then q(x)
 x in S, such that p(x) and not q(x)
o Contrapositive: has same truth table as its conditional
Contrapositive
p  q 
 ~ q  ~ p
Both do not have
Converse
 q  p to be true when
o Converse: p  q 
Inverse
o Inverse: p  q  ~ p  ~ q the original
conditional is
ture
o Biconditional: p iff q
(if p then q) and (if q then p)
p  q and q p
p q
 Problems
Page Examples
1 25
41
1,3,14
 Work Day
 Ms 1.4 to 1.5
9
10
 Lesson 1.6 Valid Arguments
o Argument: sequence of statements
o Premises: the statements of an argument
o Conclusion: final statement (usually use therefore)
 therefore
o Valid: argument form which is independent of the truth of
the statements (valid = always true)
o Law of detachment: if p then q
Use Truth Tables
p
to show that the
q
if -then argument
o Law of Transitivity: if p then q
results in all
if q then r
TRUE values
if p then r
o Law of Indirect Reasoning: if p then q
~q
 Problems
Page Examples
 ~p
1 23
49
1,4-6,14,15
 Lesson 1.7 Invalid Arguments
o Invalid: (not valid) the premises are true but the
conclusion is false
o Converse Error: if p then q
Converse
p  q 
 q  p
q
p
Inverse
o Inverse Error: if p then q
p

q

 ~ p ~ q
~p
 ~q
o Improper Induction: when tre for a few is used to imply
true for all values
o Conjecture: Statement believed true but not yet proven
 Problems
Page Examples
1 26
56
4,5,11
11
 Work Day
12
 Lesson 1.8 Direct Proofs
o Math Proof: chain of logically valid deductions using
agreed upon assumptions, definitions, or previously
proved statements
o Justifications: generalizations used in a proof
o Direct Proof: go from antecedent straight to conclusion
 Problems
Page Examples
1 23
64
13
 Ms 1.6 to 1.8
14
15


Chapter 1 Review
o Worksheet
Test Chapter 1
o Test corrections are due two days later