Although you are welcome to typeset your work nicely (using Microsoft Word
or LaTeX or whatever to get nice logic symbols),
it's probably much easier to write formulas by hand.
(Trouble reading some symbols on this page?
Here's a screen shot.)
TeachLogic exercises I: #21 (algebra).
You may justify each step by either using the
rules named as ,
or as in Rosen's §1.2 tables 6,7,8.
(Your answers will be in the same format as
the problems in
Rosen section 1.2, examples 5,6,7,8
and
TeachLogic: propositional algebra.
(Web site currently down; this problem won't be on the exam.)
TeachLogic exercises I: #9 (ebay queries).
border -common (foreign,foriegn)pilaf (superb,bowl)
As of 2007.Sep.11 15:20:
eBayQuery#((pilaf,superb bowl)) = 2
eBayQuery#((pilaf,bowl superb)) = 7
eBayQuery#(pilaf) = 2
eBayQuery#(superb bowl) = 42
eBayQuery#(bowl superb) = 42
By distributing the "or" over the "and", we get CNF:
eBayQuery#((pilaf,superb) (pilaf,bowl)) = 43 ?!
Rosen p.46, #5 (5ed: p.40,#5):
Let P(x) be the statement
x spends more than 5 hours every weekday in class,
where the domain consists of Radford students.
Express each of these quantifications in English.
∃x . P(x)
Some RU student spends ≥5hrs/weekday in class∀x . P(x)
Every RU student spends ≥5hrs/weekday in class ∃x . ¬ P(x)
Some RU student spends <5hrs/weekday in class ∀x . ¬ P(x)
No RU student spends ≥5hrs/weekday in class
Rosen p.40 #12:
Let Q(x) be the statement x+1 > 2x.
If the domain is the set of integers
Z = {…,-2,-1,0,+1,+2,…},
what are these truth values?
Q(0)
Q(-1)
Q(1)
∃x . Q(x)
∀x . Q(x)
∃x . ¬Q(x)
∀x . ¬Q(x)
Rosen p. 51, #1
Translate these statements into English:
∀x . ∃y . (x < y)
Every number has a number bigger than it,
or there is no maximum numberThat's not quite perfect;
rather
there is no maximal numberquote> ∀x . ∀y . (((x≥0) ∧ (y≥0) ) → (x·y ≥ 0))
The product of two positive numbers is positive. ∀x . ∀y . ∃ z . (x·y = z)
The product of two numbers always exists., or multiplication is defined for all inputs.Not in book:
∀x . ∀y . ∃ z . (x/y = z)
The division of two numbers always exists., or Division is defined for all inputs.
For each of the four statements above, is it true when the
domainRosen calls the domain the universe of discourse is all real numbers ℜ?
How about when the universe of discourse is restricted to the natural numbers
N = {0,1,2,3,…}?
In both cases, we'll use the standard interpretation for
the multiplication, division,
and the relations <, ≥, and multiplication.N: true
ℜ: true
N: true
ℜ: true
N: true
ℜ: true
N: false; if y=0, no number is equal to x/y.
ℜ: false; if y=0, no number is equal to x/y.
Rosen #26.
Let Q(x, y) be the statement x+y = x-y.
If the domain is the set of integers Z = {…,-2,-1,0,+1,+2,…},
what are the truth values?
Q(1, 1)
&false;
Q(2, 0)
&true;
∀y . Q(1, y)
&false; (Try y=1. Actually, any non-zero witness will do disprove this statement.)
∀x . Q(x, 2)
&false;
∃x . ∃y . Q(x,y)
&true; (Try x=0,y=0 is one solution. Any assignment with y=0 works.)
∀x . ∃y . Q(x,y)
&true; (y=0 works with any x.)
∃y. ∀x . Q(x, y)
&true; (y=0 works with any x. However, this is
actually a stronger statement than the previous, for most statements Q.)
∀y . ∃x . Q(x,y)
&false;. When y=1, there is no x which will satisfy Q(x,y).
∀x . ∀y . Q(x,y)
&false;: As seen, x=y=1 makes Q(x,y) false.
If you see a few other problems in Rosen which catch your eye,
and you'd like to do them for extra credit, you are welcome to
(though you can ask me for how much; extra-credit is harder to earn
point-per-point than regular credit).