MATHEMATICS COLLEGE

Let S(x, y) denote the statement "x has seen y" and D denote the set of all students in our class and M be the set of all movies.a. Express the following English sentence as a quantified proposition using the definitions above:
"For every movie there is a pair of students in our class who have both seen it." Hint: Use three quantifiers - one for the movie and two for a pair of students.

b. Negate the quantified proposition you wrote for part (a) (i.e., place a "Â¬" in front of it). Use de Morganâs law for quantifiers to move the negation inside the quantifiers.
c. Translate you answer for part (b) back to plain English.

Answers

Answer 1

Answer:

Answer:

Step-by-step explanation:

a) Recall the quantifiers $\forall, \exists$ .

Then, we can translate the proposition as follows

$\forall m \in M \exists x \in D \exists y \in D S(x,m)\land S(y,m)$

b) Recall that an expression of the form $\exists x P(x)$ its negation is of the form $\forall x \neg P(x)$ which means that it is not true that for all elements the proposition P holds. Equivalently, we have that the negation of an expression of the form $\forall x P(x)$ is $\exists x \neg P(x)$ which means that there is at least one x such that P doesn't hold. Using this, we get the following

$\neg(\forall m \in M \exists x \in D \exists y \in D S(x,m)\land S(y,m))= \exists m \in M \neg (\exists x \in D \exists y \in D S(x,m)\land S(y,m))= \exists m \in M \forall x \in D \neg (\exists y \in D S(x,m)\land S(y,m))= \exists m \in M \forall x \in D \forall y \in D \neg(S(x,m)\land S(y,m))$

By De Morgan's law, we have that $\neg (A \land B) = \neg A \lor \neg B$

So, the final statement is

$\exists m \in M \forall x \in D \forall y \in D \neg S(x,m) \lor \neg S(y,m)$

This statement means: There is a movie that for every pair of students, at least one of the students hasn't seen the movie yet.

Answer 2

Answer:

Final answer:

The quantified proposition expresses that for every movie, there is a pair of students who have both seen it. The negated proposition states that there exists a movie that no pair of students have both seen.

Explanation:

a. The English sentence can be expressed as the quantified proposition: ∀m ∈ M, ∃ (s1, s2) ∈ D^2, S(s1,m) ∧ S(s2,m)
b. The negation of the quantified proposition is: ∃m ∈ M, ∀(s1, s2) ∈ D^2, ¬(S(s1,m) ∧ S(s2,m))
c. The translation of the negated proposition is: There exists a movie that no pair of students in our class have both seen.

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Step-by-step explanation:

i did the test ty hope it helps and give brainly and a thanks and 5star please :D you and you will get a A on that test or homework. IT really works

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Value-?

Answers

The expression 3x + 7x - 28 + 31 - 8x simplifies to 2x + 3. When x equals 2043, the value of the Simplifying Expression is 4089.

The given expression is 3x + 7x - 28 + 31 - 8x.

To simplify the expression, we first group together the like terms.

Thus, the expression becomes (3x + 7x - 8x) + (31 - 28).

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Answer:

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Step-by-step explanation:

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