CPSC 121
Lecture 3
January 9, 2009
Menu January 9, 2009
Topics:
Conditional Statement
— Implication
— Biconditional
Logical Equivalence
Reading:
Today:
Epp 1.2
January 12: Epp 1.5
Next:
Epp 1.3
Reminders:
Labs and tutorials begin week of January 12
Online Quiz 3 deadline 9:00pm January 13
Online Quiz 2 deadline was 9:00pm January 8
Online/static Quiz 1 (mark has been waived)
WebCT Vista:
http://www.vista.ubc.ca
www:
http://www.ugrad.cs.ubc.ca/~cs121/
The first three slides for today “carry forward” from Lecture 2.
The first summarizes the role that truth tables play in propositional logic.
Why Truth Tables?
We use truth tables to:
1. Define the meaning of logical connectives
2. Construct (simple) proofs in propositional logic
3. Design (simple) combinational circuits
The second exhaustively lists the sixteen (16) different truth tables involving two propositional variables, p and q. It
also gives the name of the associated logic gate (in hardware), when there is one.
All 16 Logical Operators (with “Gate” Names)
B
*
B
X
N
N
N
U
*
U
N
A
A
X
O
O
N
O
F
*
F
O
N
N
O
T
T
O
p
q
R
p
*
q
R
D
D
R
q
p
R
T
T
T
T
T
T
T
T
T
T
F
F
F
F
F
F
F
F
T
F
T
T
T
T
F
F
F
F
T
T
T
T
F
F
F
F
F
T
T
T
F
F
T
T
F
F
T
T
F
F
T
T
F
F
F
F
T
F
T
F
T
F
T
F
T
F
T
F
T
F
T
F
*
The operator whose truth values are given in this column is called implication. Implication is very important in
formal logic (and we’ll return to it shortly). Note, however, that implication is not a named gate (in hardware)
Finally, it is important to realize that using truth tables as a proof method in propositional logic does not scale up as
the number of propositional variables increases.
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(Harder) Question
How many distinct Boolean operators are there of
n
Boolean variables?
Answer: There are
2
(2
n
)
The truth table has
2
n
rows since there are that number of distinct combinations of truth values for the
n
variables. For
each row, we have an entry that can be either true or false. Thus, there are
2
(2
n
)
distinct truth tables for
n
Boolean variables
We use truth tables as a proof method for propositional logic. But, this is feasible only when the number of variables
is small. There are other proof methods for propositional logic that we will need to consider
Recall from Last Lecture
Proposition:
a statement that is true or false (but not both)
Compound Proposition:
formed by combining existing propositions with
logical connectives
(
∧ ∨ ¬ ⊕
)
defined, via
a
truth table
, as follows:
p
q
p
∧
q
p
∨
q
¬
p
p
⊕
q
T
T
T
T
F
F
T
F
F
T
F
T
F
T
F
T
T
T
F
F
F
F
T
F
Digital circuit design uses
logic gates (NOT AND OR XOR NAND NOR XNOR)
to describe a circuit
Digital circuits produce the correct binary output for each combination of binary inputs
Question Follow Up
There is no required “correct” ordering to rows in a truth table. For example, the two truth tables
p
q
p
∧
q
p
∨
q
¬
p
p
⊕
q
T
T
T
T
F
F
T
F
F
T
F
T
F
T
F
T
T
T
F
F
F
F
T
F
p
q
AND
OR
NOT p
XOR
0
0
0
0
1
0
0
1
0
1
1
1
1
0
0
1
0
1
1
1
1
1
0
0
are equivalent (when we associate T with 1 and F with 0)
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 Spring '08
 BELLEVILLE
 Logic, Logical connective

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