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hw02
hw02
Due 2007.Sep.06 (Thu) noon
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.)
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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.
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TeachLogic exercises I: #9 (ebay queries).
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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)
- ∀x . P(x)
- ∃x . ¬ P(x)
- ∀x . ¬ P(x)
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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?
- true or false?: Q(0)
- true or false?: Q(-1)
- true or false?: Q(1)
- true or false?: ∃x . Q(x)
- true or false?: ∀x . Q(x)
- true or false?: ∃x . ¬Q(x)
- true or false?: ∀x . ¬Q(x)
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Rosen p. 51, #1
Translate these statements into English:
- ∀x . ∃y . (x < y)
- ∀x . ∀y . (((x≥0) ∧ (y≥0) ) → (x·y ≥ 0))
- ∀x . ∀y . ∃ z . (x·y = z)
- Not in book:
∀x . ∀y . ∃ z . (x/y = z)
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For each of the four statements above, is it true when the
domain1 is all real numbers ℜ?
How about when the universe of discourse is restricted to the natural numbers
N = {0,1,2,3,…}?
2
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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?
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Q(1, 1)
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Q(2, 0)
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∀y . Q(1, y)
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∀x . Q(x, 2)
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∃x . ∃y . Q(x,y)
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∀x . ∃y . Q(x,y)
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∃y. ∀x . Q(x, y)
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∀y . ∃x . Q(x,y)
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∀x . ∀y . Q(x,y)
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).
1Rosen calls the domain “the universe of discourse” ↩
2In both cases, we'll use the standard interpretation for
the multiplication, division,
and the relations <, ≥, and multiplication. ↩
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