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Let the universe of discourse U

= all books and people. Let B be the set of all books and P be the set of all people. In the questions below, assume b ∈ B and p ∈ P. Let R(p,b) = p has read b.

Translate the following:

(a) R(Sarah,Moby Dick)

(b) ∃p R(p,Moby Dick)

(c) ∀b ¬R(Sarah,b)

(d) ∃b R(Sarah,b)

(e) ∀p ¬R(p,Moby Dick)

(f) ¬∀p∃b R(p,b)

(g) ∃p∀b R(p,b)

(h) ∃p∃b R(p,b)

2. Let the universe of discourse U = all people. Let F(x,y) = x follows y on Instagram.

Translate the following:

(i) ∀x∃y F(x,y)

(j) ∃x∀y F(x,y)

(k) ∀x∃y F(y,x)

(l) ∃x∀y F(y,x)

(m) Someone follows everyone on Instagram.

(n) There is someone whom everyone follows on Instagram.

(o) Everyone follows someone on Instagram.

(p) Everyone has an Instagram follower.

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3. Let the universe of discourse U= all students writing this homework. Given the following predicates, translate the following statements from predicate logic to English. Try your best to translate into colloquial English rather than its literal translation.

F(x) = x is having fun.

S(x) = x is feeling sleepy.

C(x) = x is drinking coffee.

For example, ∀xF(x) is translated to “everyone writing this homework is having fun.”

(q) ∀x C(x)

(r) ∃x (S(x) ∧ C(x))

(s) ¬∀x F(x)

(t) ¬∃x ¬F(x)

(u) ∀x (S(x) −→ C(x))

(v) ∀x (¬F(x) −→ S(x))

(w) ¬∃x (F(x) ∧ S(x))

4. Let the universe of discourse U= all internship applications. Given the following predicates, translate the following English statements into predicate logic.

I(x) = x is important.

R(x) = x is reviewed.

A(x) = x is accepted.

(x) Every internship application is important.

(y) All internship applications are important and reviewed.

(z) Some internship applications are not reviewed.

(aa) Not all internship applications are reviewed or accepted.

(ab) Only some internship applications are accepted.

(ac) Not all important internship applications are accepted.

(ad) All internship applications that are accepted have been reviewed.

(ae) Not all internship applications that are important and reviewed are accepted.

Let the universe of discourse U= all people. Let Y (x) be the predicate “x has a

YouTube channel”. For each of the statements below, do the following:

(af) Translate the both statements into English.

(ag) Use De Morgan’s Law on the second statement and translate into English.

(ah) Determine if the two statements have the same meaning.

Statement 1 : ∃x ¬Y (x).

Statement 2 : ¬∀x Y (x).

Statement 1 : ∀x Y (x).

Statement 2 : ¬∀x ¬Y (x).

Let the universe of discourse U

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Bonus (5pt) Let the universe of discourse U = all people. Let the predicate O(x,y) = x has shook hands with y. Below is an incorrect translation for the statement “everyone has shook hands with everyone else.”

∀x∀y O(x,y)

What is wrong with this translation? What can you do to fix this translation?

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