Discreet Math
dunpel1
DiscreetMathweek1.docxLet the following statements be given.
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"You are in Seoul." |
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"You are in Kwangju." |
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"You are in South Korea." |
Translate the following statement into symbols of formal logic.
If you are not in South Korea, then you are not in Seoul or Kwangju.
¬r → ¬(p ∧ q)
r → ¬(p ∨ q)
¬p → ¬q
¬r → ¬(p ∨ q)
r → ¬(p ∧ q)
2. –/3 points HunterDM2 1.1.010a. My Notes
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Use truth tables to establish the following equivalence. Show that
¬(p ∨ q)
is logically equivalent to
¬p ∧ ¬q.
This equivalence is one of De Morgan's laws, after the nineteenth-century logician Augustus De Morgan.
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p ∨ q |
¬(p ∨ q) |
¬p ∧ ¬q |
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3. –/3 points HunterDM2 1.3.006b. My Notes
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The domain of the following predicates is the set of all plants.
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"x is poisonous." |
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"Jeff has eaten x." |
Translate the following statement into predicate logic.
Jeff has never eaten a poisonous plant.
(∀x)(Q(x) → P(x))
(∀x)(P(x) → ¬Q(x))
(∃x)(¬Q(x) → P(x))
(∃x)(¬P(x) ∧ ¬Q(x))
(∃x)(P(x) → Q(x))
4. –/3 points HunterDM2 1.2.008. My Notes
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Which derivation rule justifies the following argument?
If n is a multiple of 4, then n is even. However, n is not even. Therefore, n is not a multiple of 4.
double negation modus ponens simplification De Morgan's Laws modus tollens
5. –/3 points HunterDM2 1.2.018. My Notes
Write a proof sequence for the following assertion. Justify one of the steps in your proof using the result of Example 1.8.
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(No Response)
This answer has not been graded yet.
6. –/3 points HunterDM2 1.2.024. My Notes
Use a truth table to show that
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is not a tautology. (This example shows that substitution isn't valid for inference rules, in general. Substituting the weaker statement, q, for the stronger statement, p, in the expression "¬p" doesn't work.)
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p → q |
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(p → q) ∧ ¬p |
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((p → q) ∧ ¬p) → ¬q |
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7. –/3 points HunterDM2 1.1.026. My Notes
Often a complicated expression in formal logic can be simplified. For example, consider the statement
S = (p ∧ q) ∨ (p ∧ ¬q).
(a) Construct a truth table for S.
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p ∧ q |
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p ∧ ¬q |
S |
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(b) Find a simpler expression that is logically equivalent to S.
p p ∧ q ¬(p ∧ q) q ¬(p ∨ ¬q)
8. –/3 points HunterDM2 1.3.022aiii. My Notes
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Let the following predicates be given. The domain is all cars.
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"x is fast." |
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"x is a sports car." |
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"x is expensive." |
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"x is safer than y." |
Write the following statement in predicate logic.
Every fast sports car is expensive.
(∀x)((F(x) ∧ S(x)) → E(x))
(∀x)(F(x) ∧ S(x))
(∀x)((F(x) ∨ S(x)) → E(x))
(∃x)((E(x) ∧ S(x)) → F(x))
(∀x)((E(x) ∨ S(x)) → ¬F(x))
9. –/3 points HunterDM2 1.3.004a. My Notes
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Let the following predicates be given. The domain is all mammals.
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"x is a lion." |
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F(x) |
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"x is fuzzy." |
Translate the following statement into predicate logic.
All lions are fuzzy.
(∀x)(F(x) → ¬L(x))
(∀x)(F(x) → L(x))
(∀x)(¬L(x) → F(x))
(∃x)(L(x) ∧ F(x))
(∀x)(L(x) → F(x))
10. –/3 points HunterDM2 1.3.002d. My Notes
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In the domain of all penguins, let D(x) be the predicate "x is dangerous." Translate the following quantified statement into simple, everyday English.
(∃x)¬D(x)
No penguins are dangerous. Some penguins are dangerous. Some penguins are dangerous while other penguins are not dangerous. Some penguins are not dangerous. All penguins are dangerous.
