Brain Teaser
You are speaking to twins, Joe and Jim. One
always tells the truth and one always lies, but
you don’t know which is which. You ask one
twin, “Does Joe always lie?”. He answers, “Yes”.
Did you speak to Joe or Jim?
Which twin is the liar?
Some Basics •
•
•
•
Number sequences
Propositions
Truth tables
Boolean variables
Number Sequences
•
•
•
•
5,
1,
1,
2,
7,
9,
4,
4,
9, 11, 13, ?, …
17, 25, 33, 41, ?, …
9, 16, 25, ?, …
8, 16, 32, ?, …
Number Sequences
•  Recursive formula: each term is specified
by arithmetic operations on previous
term(s). Must initialize.
–  Recurrence relation
•  Closed formula: each term specified in
terms (arithmetic) of its position in
sequence
a1 , a2 , a3 , . . .
Two sided
Nth element
an
. . . a−2 , a−1 , a0 , a1 , a2 , . . .
Number Sequences
•  Sequence notation
a0 , a1 , a2 , . . .
Two sided
Nth element
an
. . . a−2 , a−1 , a0 , a1 , a2 , . . .
•  Notation for sums
n
!
k=1
ak = a1 + a2 + a3 + . . . + an
Practice Problem
•  Give recursive and closed form
expressions for the following:
a)
b)
c)
d)
5,
1,
1,
2,
7,
9,
4,
4,
9, 11, 13, …
17, 25, 33, 41, …
9, 16, 25, …
8, 16, 32, …
Propositions
•  A proposition is a statement that can be
either true or false.
–  “Today is Wednesday.”
–  “I love CS 2100.”
–  “x = 7”
•  Not propositions:
–  “Why am I here?”
–  “Read the book.”
Boolean Variables
•  We’ll use variables (letters) to represent
propositions.
–  p = “today is Wednesday”
–  q = “it is raining”
•  A variable may take one of two values:
true (T) or false (F).
“Boolean”?
In 1854 he published An Investigation into
the Laws of Thought, on Which are
founded the Mathematical Theories of
Logic and Probabilities. Boole approached
logic in a new way, reducing it to a
simple algebra incorporating logic into
mathematics. He pointed out the analogy
between algebraic symbols and those that
represent logical forms. It began the
algebra of logic called Boolean algebra
which now finds application in computer
construction, switching circuits, etc.
Truth Tables
•  List the possibilities of boolean variables
•  Game: truth-tellers and liars
•  Scenario: meet person A and B
–  A: “Exactly one of us is lying”
–  B: “At least one of us is telling the truth”
•  Goal: Identify A and B as truth-tellers
or liars
Truth Tables
•  Introduce propositions:
–  p
–  q
–  r
–  s
–
–
–
–
A is a truth teller
B is a truth teller
A’s statement
B’s statement
Truth Tables
Truth Tables
Find scenario with no contradictions
Paradoxes
•  “I am telling the truth”
•  “I am lying”
Joe and Jim
You are speaking to twins, Joe and Jim. One
always tells the truth and one always lies, but
you don’t know which is which. You ask one
twin, “Does Joe always lie?”. He answers, “Yes”.
Did you speak to Joe or Jim?
Which twin is the liar?
Joe and Jim
•  Introduce propositions:
–  p –
–  q –
–  Cases
–  A –
–  B –
Joe is a truth teller
Jim is a truth teller
We spoke with Jim
We spoke with Joe
Try to Fill out Truth Tables •  Look for inconsistencies/consistencies
•  Case A:
•  Case B:
Logical Operators
•  Logical operators operate on Boolean
variables, much like algebraic operators
(+ , - , * , / ) operate on numerical
variables
•  Just like propositions, operators can be
converted into words: “and”, “or”, “not”
And (∧)
•  An “and” operation is true if
both arguments are true
•  In C++ and Java the
operator is &&
•  Example:
p ∧ q = “Today is Wednesday
and it is raining”
Or (∨)
•  An “or” operation is true
if at least one argument
is true
•  In C++ and Java the
operator is ||
•  Example:
p ∨ q = “Today is Wednesday
or it is raining (or both)”
Not (¬)
•  A “not” operation negates
(flips) its argument
•  In C++ and Java the
operator is !
•  Example:
¬p = “Today is not Wednesday”
Switching Circuits/Digital Logic
Truth Tables for Formal
Propositions
•  Evaluate
•  Work inside out (hierarchical)
–  Use precedence rules and parentheses
Logical equivalence
•  Two statements are said to be logically
equivalent if they have the same value
for every row of a truth table
•  DeMorgan’s Law
Logical Constants
•  A tautology is always true
•  A contradiction is always false
•  Example: prove the following is a
tautology
Properties of Boolean Operations