Discrete Structures

# Lecture Notes 9/8/10

 Last Update: 8 September 2010, 9:11 P.M. Note: or material is highlighted

1. First, here's the complete chart, including the old stuff from last time & the new material from today:

Syntax & Semantics of Our Language for Propositional Logic

Given propositions p & q, we can generate (or construct) more complex propositions.

NAME SYNTAX ENGLISH E.G. SEMANTICS
atomic
proposition
p, q, etc. (any "simple" grammatical
declarative sentence)
Today is Monday.
2+2=4
tval(p)=T
iff
p describes the world correctly
molecular
propositions:

negation

¬p It's not the case that p.
Not p
Today isn't Monday.
2+2≠4
Use truth tables
for the semantics of
molecular propositions:

INPUTOUTPUT
p¬p
TF
FT

i.e., tval(¬p)=F iff tval(p)=T
& tval(¬p)=T iff tval(p)=F

conjunction (pq) p and q
p but q
p although q
etc.
Today is Monday and 2+2=4.
Today is Monday but there's no school anyway.
I/PO/P
pq(p ∧ q)
TTT
TFF
FTF
FFF
inclusive
disjunction

(Latin "vel")

(pq) (Either) p or q (or both)
p and/or q
Today is Monday or 2+2=4.

You'll pass if you study or if you already know the material.

I/PO/P
pq(p ∨ q)
TTT
TFT
FTT
FFF
exclusive
disjunction

(Latin "aut")

(pq) (Either) p or (else) q (but not both) Today is a weekday or a weekend.
You'll pass 191 or else you'll fail 191.
I/PO/P
pq(p ⊕ q)
TTF
TFT
FTT
FFF

Note: tval(pq)=T iff tval(p)≠tval(q)
i.e., T iff different tvals.

material
conditional
(pq)

p is the "antecedent"
(= "goes before")

q is the "consequent"
(= "follows together")

If p, then q
q if p
p only if q
p is a sufficient condition for q
q is a necessary condition for p
If today is Tuesday, this must be Belgium.
(title of famous movie)

If the sum of the digits of a number is divisible by 3,
then so is the number.

I/PO/P
pq(p → q)
TTT (easy)
TFF (obvious)
FTT (!)
FFT (!)

For explanations, see here and here.

biconditional (pq) p if & only if q
p iff q
p is necessary & sufficient for q
q is necessary & sufficient for p
A plane figure is a triangle iff it is a 3-sided polygon.
I/PO/P
pq(p ↔ q)
TTT
TFF
FTF
FFT

Note: tval(pq)=T iff tval(p)=tval(q)
i.e., T iff same tvals.
i.e., (pq) is the negation of (pq)!
(later, we'll have a precise way of saying that)

2. How to compute a truth table for a molecular proposition:

1. Consider this molecular proposition:

((pq) ∧ ¬(pq))

2. It will have 4 possible truth values corresponding to the 4 possible combinations of truth values of its atomic propositions pq

3. But we don't want to show just the 2 "input" columns of possible truth values for the atomic propositions
and the "output" column of the truth values for the molecular proposition.

4. Instead, we want to show how the output can be computed from the input (after all, we're computer scientists!).

5. To do that, we need to show the "intermediate" columns where all the work is done.

6. There will be one intermediate column for each "constituent" part of the molecular proposition.

7. Our example molecular proposition is basically a conjunction of the form (A ∧ B), where:

1. A is basically a disjunction of the form (p ∨ q), where:

1. p and q are atomic propositions

2. and B is basically a negation of the form ¬C, where:

1. C is basically a conjunction of the form (pq), where:

• p and q are atomic propositions
(in fact, the same ones as before)

8. So, here is the full truth table, with our 2 input columns of atomic propositions, our 3 intermediate columns of "constituent" parts of the molecular proposition, and our output column for the molecular proposition:

input intermediate output
12 3=(1∨2)4=(1∧2)5=¬4 6=(3∧5)
pq (pq) (pq) ¬(pq) ((pq) ∧ ¬(pq))
TTTTFF
TFTFTT
FTTFTT
FFFF TF

9. Note that our molecular proposition has the same truth table as exclusive or: (pq)

• We'll come back to this later.
Next lecture…

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