1.(3 points) Let L(x) denote the statement “x has used LaTeX.” and D denote the
domain of all students in our class. Express each of these quantified propositions in
English:
a.∀x∈D : L(x)
Answer:
Each student in our class has used LaTeX.
b.¬9x2D : L(x)
Answer:
It isn't the case that some students in our class has used LaTeX.
c.∃x∈D : ¬L(x)
Answer:
There are some students in our class who has not used LaTeX
2.(2 points) Let Q(x) denote the statement “2x > 3x” and Z denote all integers (i.e., . . . ,
−3, −2, −1, 0, 1, 2, 3, . . .). Determine the truth value of each of these statements:
(a)Q (2)
Answer:
2
2
> 2 * 3 <=> 4 > 6
So, it’s False
(b)Q (4)
Answer:
2
4
> 4 * 3 <=>16 > 12
So, it’s True
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(c)∀x∈Z : (Q(x) ∨ x < 4)
Answer:
Evidently the proposition is true for x < 4. In the above answer, we saw that, Q(x)
is true when x = 4. This remains true as x increases.
So, it’s True
(d)∃x∈Z : (Q(x) ∧ x < 4)
Answer:
Let’s choose x = 0. Evidently 0 < 4 and 2
> 3 * 0 <=> 1 > 0.
So, it’s True
3.(2 points) Translate the following statements to English where B(x) is “x understands
boolean algebra” and M(x) is “x has taken discrete math” and the domain D is all
students at UTSA.
(a)∀x∈D : (M(x) → B(x))
Answer:
For all students at UTSA, if the student has taken discrete math, then they
understand boolean algebra.
(b)∃x∈D : (B(x) ∧ ¬M(x))
Answer:
There is a student at UTSA in that this student understands boolean algebra yet
has not taken discrete math.
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CS2233 Discrete Mathematical StructuresHomework 21. Predicates and Quantifiers (7 points)1.(3 points) Let L(x) denote the statement “x has used LaTeX.” and D denote thedomain of all students in our class. Express each of these quantified propositions inEnglish:a. ∀x∈D : L(x)Answer:Each student in our class has used LaTeX.b. ¬9x2D : L(x)Answer:It isn't the case that some students in our class has used LaTeX.c. ∃x∈D : ¬L(x)Answer:There are some students in our class who has not used LaTeX2. (2 points) Let Q(x) denote the statement “2x > 3x” and Z denote all integers (i.e., . . . ,−3, −2, −1, 0, 1, 2, 3, . . .). Determine the truth value of each of these statements:(a) Q (2)Answer:22 > 2 * 3 4 > 6So, it’s False(b) Q (4)Answer:24 > 4 * 3 16 > 12So, it’s True(c) ∀x∈Z : (Q(x) ∨ x < 4)Answer:Evidently the proposition is true for x < 4. In the above answer, we saw that, Q(x)is true when x = 4. This remains true as x increases.So, it’s True(d) ∃x∈Z : (Q(x) ∧ x < 4)Answer:Let’s choose x = 0. Evidently 0 < 4 and 20 > 3 * 0 1 > 0.So, it’s True3. (2 points) Translate the following statements to English where B(x) is “x understandsboolean algebra” and M(x) is “x has taken discrete math” and the domain D is allstudents at UTSA.(a) ∀x∈D : (M(x) → B(x))Answer:For all students at UTSA, if the student has taken discrete math, then theyunderstand boolean algebra.(b) ∃x∈D : ... Purchase document to see full attachment
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