1.1 Sets and Logic
• Set – a collection of objects. Set brackets {} are
used to enclose the elements of a set.
Example: {1, 2, 5, 9}
• Elements – objects inside the brackets
2  A means 2 is an element of set A
3  A means 3 is not an element of set A
• Cardinal number – number of elements of a set
notation: n(A) = # elements in set A
1.1 Sets and Logic
• Sets are equal – they contain the same
elements (the order can be different)
example: {A, B, C} = {B, C, A}
• {x | x has the property y} – This is read:
“The set of x such that x has the property y”
examples: {x | x is a letter grade}
{x | x is an integer between –1.5 and 5.2}
1.1 Sets and Logic
• Universal set – set of all elements in a given
situation
example: all outcomes when a die is rolled
U = {1, 2, 3, 4, 5, 6}
• Empty set – set of no elements, denoted by 
• Subset – B  A (B is a subset of A) true if every
element of B is also an element of A
• Proper subset – B  A (B is a proper subset of A)
true if B  A and B  A
1.1 Sets and Logic
• For all sets:
  A and A  A
• # of subsets – a set with n distinct elements
has 2n subsets
• {} is different from ;  = {}
 has no elements (cardinality = 0)
{} has one element (cardinality = 1)
1.1 Sets and Logic
• Pascal’s triangle can be used to find the number
of subsets with a given number of elements.
1
1
1
1
1
1
1
1
1
1
8
9
10
35
1
5
15
35
70
126
1
4
20
56
84
3
10
21
1
6
15
28
36
3
5
7
2
4
6
1
6
21
56
126
1
1
7
28
84
1
8
36
1
9
1
1.2 Set Operations
• Complement of a set A – the set of all elements
that are in the universal set associated with set A
but not in A itself.
In text: AC = complement of A
example: U = {1, 2, 3, 4, 5, 6} A= {1, 2}
then AC = {3, 4, 5, 6}
• Cardinalities: n(A) + n(AC) = n(U)
example: n(U) = 12 and n(A) = 3; find n(AC)
n(AC) = n(U) – n(A) = 12 – 3 = 9
1.2 Set Operations
• Venn diagrams – useful for visualizing sets
AC
A
A
A set and its complement
BA
B
1.2 Set Operations
• General Venn diagram for 2 sets
A
II
If A  B region II is empty
B
III
If B  A region IV is empty
IV
I
1.2 Set Operations
• Union – The union of two sets A & B is the set
that contains all the elements that are in A or B or
both A and B – denoted AB
(regions II, III, and IV above)
• Intersection – The set of all elements that are in
both A and B – denoted by AB
(region III above)
• Disjoint sets – If 2 sets have no elements in
common they are disjoint - AB = 
(region III is empty)
1.3 Sets and Venn Diagrams
• De Morgan’s Laws for sets:
– ACBC = (AB)C
– ACBC = (AB)C
1. 3 Sets and Venn Diagrams
• General Venn diagram for 3 sets
A
B
C
Divided into 8 regions
1.3 Sets and Venn Diagrams
• Venn diagram - shading
A
B
A  B: crisscross area
A  B: all shaded area
1.3 Sets and Venn Diagrams
• Venn diagram – disjoint sets
A B 
A
B
1.3 Sets and Venn Diagrams
• Cardinality rule for the union of 2 sets:
n(AB) = n(A) + n(B) - n(AB)
• Cardinality rule for the union of 3 sets:
n(ABC) =
n(A) + n(B) + n(C) - n(AB) - n(BC) - n(AC)
+ n(ABC)
1.4 Inductive and Deductive Logic
• Inductive Logic – is the process of drawing
a general conclusion from specific case.
Example: When a number ending in 5 is squared,
does the result end in 25?
52 = 25
152 = 225
252 = 625
552 = 3025
952 = 9025
1252 = 15625
Inductive logic says this is true
1.4 Inductive and Deductive Logic
• Inductive logic sometimes gives you a false
conclusion.
Example: Does the expression n2 – n + 11 always
give a prime number?
For n=2, n2 – n + 11 = 13 prime
For n=3, n2 – n + 11 = 17 prime
For n=4, n2 – n + 11 = 23 prime
For n=5, n2 – n + 11 = 31 prime
For n=6, n2 – n + 11 = 41 prime
1.4 Inductive and Deductive Logic
• Example: Does the expression n2 – n + 11
always give a prime number?
For n=7, n2 – n + 11 = 53 prime
For n=8, n2 – n + 11 = 67 prime
For n=9, n2 – n + 11 = 83 prime
For n=10, n2 – n + 11 = 101 prime
For n=11, n2 – n + 11 = 121 = 112 not prime
Finally we get a counterexample!
1.4 Inductive and Deductive Logic
• Counterexample – a single case or example
that is used to refute a mathematical
conjecture
• Deduction – the process of drawing a
specific conclusion from a general situation.
• Basic Syllogism (deductive logic)
– 2 statements (premises and a conclusion
1.4 Inductive and Deductive Logic
• Inductive Logic (sometimes valid)
Specific cases  general case
• Deductive logic (always valid)
General case  specific cases
1.5 Logic Statements
• Statement – sentence that has a truth value. The
statement is either true or false but not both
• Negation of a statement – a statement whose truth
value is always the opposite that of the original
statement. The negation of P is ~P.
• Quantifier – a word or phrase describing the
inclusiveness of the statement.
Examples: some, all most, few
1.5 Logic Statements
• The Accord is manufactured by Honda (statement)
• Mathematics is the best subject
(not a statement - opinion)
• Earth is the only planet in the universe (statement)
• What are fireflies? (not a statement – question)
• 2–x=3
(not a statement – equation with a variable)
• 1 = 2 (statement)
1.5 Logic Statements
Quantifier for
statement
All
Negation
Some
None
None
At least one is
At least one is not
1.5 Logic Statements
• Paradox – a statement or group of
statements that results in a contradiction
Example: “This statement is false”
- it cannot be given a truth value
• Zeno’s Paradox – Achilles and the tortoise
(on page 34 of text)
1.6 Compound Statements
• Definition: A truth table for a statement is a table
that provides the truth value of the statement for
all possible situations
• Definition: Two statements are logically
equivalent if they have the same truth tables
• Definition: Conjunction of two statements p and q
is the statement “p and q” – which is only true if
both p and q are true. Notation: p  q
1.6 Compound Statements
• Definition: Disjunction of two statements p and q
is the statement “p or q” – which is true if either p
or q are true. Notation: p  q
• Truth Tables:
p
q
pq
p q
T
T
T
T
T
F
T
F
F
T
T
F
F
F
F
F
1.6 Compound Statements
• De Morgan’s Laws for negation:
– ~(p  q) = (~p)  (~q)
– ~(p  q) = (~p)  (~q)
1.7 Conditional Statements
• Conditional statement - can be put in the form “if
p then q” (Notation: pq)
• P is the antecedent or hypothesis; Q is the
consequent or conclusion
p
q
pq
• Truth table:
T
T
T
T
F
F
F
T
F
F
T
T
1.7 Conditional Statements
• Ways to translate pq:
– If p then q
– P only if q
– P implies q
– P is sufficient for q
– Q is necessary for p
– Q if p
– All p are q
1.7 Conditional Statements
• Tautology - A compound statement that is true
under all possible truth assignments.
example: p  ~p
• Contingency - A compound statement that is
sometimes true and sometimes false depending on
truth assignments
example: pq
• Contradiction - A compound statement” that is
false under all possible truth assignments
example: p  ~p
1.8 More Conditionals
• Converse of a conditional statement - formed by
interchanging the hypothesis and the conclusion.
example: converse of pq is qp
• Inverse of a conditional statement - formed by
negating the hypothesis and the conclusion.
example: inverse of pq is ~p~q
• Contrapositive of a conditional statement - formed
by interchanging and negating the hypothesis and
conclusion.
example: contrapositive of pq is ~q~p
1.8 More Conditionals
•
Conditional: pq Converse: qp
• Contrapositive: ~q~p Inverse: ~p~q
• Rule: Interchanging and negating the hypothesis
and conclusion gives an equivalent conditional
1.8 More Conditionals
• Biconditional statement - can be put in the form “p
if and only if q” (Notation: pq)
• Truth table:
p
T
T
F
F
q
T
F
T
F
pq
T
F
F
T
1.9 Analyzing Logical Arguments
• Definition: If pq is a tautology, then q
“logically follows” from p
• Definition: conditional representation of an
argument
is [p1 p2  p3……..  pn]q
1.9 Analyzing Logical Arguments
Direct Proof
Transitive Proof
1. pq
Proof by
contradiction
1. pq
2. p
2. ~q
2. qr
q
~p
 pr
1. pq
1.9 Analyzing Logical Arguments
• Definition: A fallacy is an argument that may seem
to be a valid logical argument, but in fact is
invalid.
a=b
ab = b2
ab – a2 = b2 – a2
a(b – a) = (b – a)(b + a)
a=b+a
a = 2a
1=2
1.9 Analyzing Logical Arguments
Fallacy:
Fallacy:
Affirming the consequent Denying the antecedent
1. pq
1. pq
2. q
2. ~p
p
~q
1.9 Analyzing Logical Arguments
• Proof – affirming the consequent is not valid
• Truth table for [(p  q)  q]  p:
p
T
T
F
F
q
T
F
T
F
pq
T
F
T
T
[(p  q)  q]
[(p  q)  q]  p
T
F
T
F
T
T
F
T
1.10 Logical Circuits
• Definition: Switch is an electronic component that
can either have power flowing through it or not.
Note: This is comparable to a logic statement
– Switch – “on” or “off”
– Statement – “T” or “F”
• Definition: A group of switches connected
together is a circuit
1.10 Logical Circuits
• Definition: “series circuit” – connection of
two or more switches so that the circuit
works only if both switches are on.
p
q
1.10 Logical Circuits
• Definition: “parallel circuit” – connection of
two or more switches so that the circuit
works if either of the switches is on.
p
q
1.10 Logical Circuits
• Definition: “complementary switches” –
switches that are set up so that when one is
on, the other is off and vice versa.
~p
1.10 Logical Circuits
• Open and closed switches:
open = false, closed = true (current flows)
p is open (false), q is closed (true)
p
q