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Papessa [141]
3 years ago
10

Let D = E = {−2, −1, 0, 1, 2} Write negations for each of the following statements and determine which is true, the given statem

ent or its negation. a. ∀x ∈ D, ∃y ∈ E such that x + y = 1. b ∃x ∈ D such that ∀y ∈ E, x + y = y. c. ∀x ∈ D, ∃y ∈ E such that xy ≥ y. d. ∃x ∈ D such that ∀y ∈ E, x ≤ y.
Mathematics
1 answer:
vfiekz [6]3 years ago
5 0

Answer:

See explanation below.

Step-by-step explanation:

We use laws of quantifiers to write the negations. The negation of ∀w∃zP(w,z) is ∃w∀z¬P(w,z). The negation of ∃w∀zP(w,z) is ∀w∃z¬P(w,z), for a predicate P depending on w and z.  

a) Negation: ∃x ∈ D, such that ∀y ∈ E, x + y ≠ 1. In this case the negation is true, hence the original statement is false. Take x=-2∈D, and note that x+y≤0 for all y ∈ E. Therefore x+y is never 1.

b) Negation: ∀x ∈ D, ∃y ∈ E such that x + y ≠ y. Here, the original statement is true. Take x=0∈D, then x+y=0+y=y for all y∈E.

c) Negation: ∃x ∈ D, such that ∀y ∈ E, xy < y. In this case, the original statement is true. Given x∈D, take y=x∈E. Hence xy=x²≥x for all x∈D (this inequality is true for integers, and we can check it case by case in D).

d) Negation: ∀x ∈ D, ∃y ∈ E such that x > y. In this case, the original statement is true, take x=-2∈D. Then -2≤y for all y∈E.

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Answer:

Yes, I agree

Step-by-step explanation:

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A cubic function is represented as:

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A cubic function may have 1, 2 or 3 x intercepts. This is shown below

For 3 x intercepts

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It has been shown above that a cubic function may have 1, 2 or 3.

So, I agree to the statement

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A quadratic function will not have any x intercept when the function can not be factorized;

E.g.

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<em>The above function has no x intercept.</em>

A quadratic function will have at least 1 x intercept when the function can be factorized;

E.g.

y = x^2- 6x + 9

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We've shown that a quadratic may have no x intercept, and it may also have x intercept(s)

Hence, I agree to both statement

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