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Logic, sets, and recursion

Robert L. Causey

Chapter 4

Predicate Calculus - all with Video Answers

Educators

Chapter Questions

Problem 1

Using $\left(\left((\forall x) P^1 x \wedge M^1 b\right) \wedge \neg(\exists y)(\forall x)\left(R^2 x y \vee S^2 y x\right)\right)$ as the root node, sketch the tree structure of this formula. This kind of tree should have a double branch for each occurrence of $\wedge, \vee, \rightarrow, \leftrightarrow$, and a single branch for each occurrence of $\neg$ and each occurrence of a quantifier. The leaves of the tree must be atomic formulas.

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05:53

Problem 2

According to the precise and strict application of the syntax rules, determine which of the following are $P C$ formulas. Rewrite the ones that are genuine formulas without using unnecessary outer parentheses and without superscripts.
1. $\left(P_3 \rightarrow R^2 x y\right)$
2. $\left(P_3^1 \rightarrow R^2 x y\right)$
3. $\left(P^1 a \wedge b\right)$
4. $\left(R^3 a x r \vee(\forall x) Q^2 x y\right)$
5. $(\forall x)(\exists y) R^2 x y$
6. $\neg(\exists z)\left(A^1 z \rightarrow\left(B^1 z \wedge \neg R^2 x y\right)\right)$
7. $\neg(\exists z)(\exists y) R^2 y x$
8. $\left((\exists x) P^1 x \rightarrow(\forall x) a\right)$
9. $(\exists y) \neg(\forall x) M^3 x b y$
10. $\left(P^1 x \rightarrow P^2 a b\right)$
11. $(\forall x)\left(M^1 x\right)$
12. $(\forall x)\left(R_4^2 x d \rightarrow(\exists y)\left(P^1 y \wedge R_4^2 q y\right)\right)$
13. $\left(R \leftrightarrow\left(R^2 x z b \vee B^1 a\right)\right)$
14. $\neg \neg \neg\left(A \vee(\exists x)(\exists y)(\exists z)\left(B^2 x u \rightarrow A^1 z\right)\right)$
15. $(\forall x y) M_1^1 x$
16. $(\forall x)(\forall y)\left(M_1^1 x \vee\left(\neg M_1^1 u \wedge M^1 y\right)\right)$

Matthew Winsor
Matthew Winsor
Numerade Educator
01:47

Problem 3

After you have completed the previous exercise, examine the formulas that you have rewritten. For each formula, specify all free occurrences of variables in it. If it has no free occurrences of any variables, indicate that it is a sentence.

Shahab Ullah
Shahab Ullah
Numerade Educator
01:00

Problem 4

Give examples of PC formulas with the following features:
1. A sentence that is a universal generalization of a conditional, in which the consequent of this conditional is an existential generalization.
2. A formula that is not a sentence but which is an existential generalization of a conjunction.
3. A formula with at least three free occurrences of exactly two different variables.
4. A sentence that is a disjunction of: an atomic sentence with a 3 -ary predicate and an existential generalization.
5. A formula that is not a sentence but which has occurrences of two universal quantifiers with different variables and one existential quantifier, and in addition becomes a sentence when it is universally generalized with one additional quantifier.

Raj Bala
Raj Bala
Numerade Educator

Problem 5

Consider a set of books in a library. Let the variables $s, t, u, \cdots$ pertain to these books, and to human beings, as we used variables in the earlier example about the bag of objects. Use predicates and constants to represent information as follows:
$\begin{array}{ll}r & : \text { The Rubáiyát of Omar Khayyám } \\ f & : \text { Les Fleurs du Mal } \\ b & : \text { Begriffschrift } \\ e & : \text { An Essay Concerning Human Understanding } \\ g & : \text { G. Frege } \\ c & : \text { Charles Baudelaire } \\ L x y & : x \text { is a longer book than } y \\ W x y & : x \text { is a book written by } y \\ F x & : x \text { is a book written in French } \\ P x & : x \text { is a philosopher } \\ M x & : x \text { is a mathematician }\end{array}$
We can now symbolize some English sentences, e.g.,
Les Fleurs du Mal is a book written in French.
Ff.
An Essay Concerning Human Understanding is (a book) written by a philosopher and is a longer book than The Rubáiyát.
$$
((\exists y)(W e y \wedge P y) \wedge \text { Ler }) .
$$
There is a book written by a philosopher and it is longer (i.e., a longer book) than The Rubáiyát.
$(\exists x)((\exists y)(W x y \wedge P y) \wedge L x r)$.
Symbolizations are not unique because any $P C$ sentence is logically equivalent to other $P C$ sentences in other forms. Also, a set of English predicates can often be symbolized into more than one set of equally satisfactory $P C$ predicates.

Using commonsense interpretations of the meanings of the following English sentences, symbolize each of them in predicate calculus. It does not matter whether the English sentences are true or false; just try to capture their meanings. Try to find $P C$ sentences (no free occurrences of variables) that have structures close to the structures of the corresponding English sentences. Symbolizations can be difficult; a good understanding of logic and much practice are required to do them in an effective and elegant way.
1. There is a book written by Baudelaire.
2. The Rubáiyát is a book written by a mathematician.
3. If The Rubáiyát is a book written by a mathematician, then it (The Rubáiyat) is a book written by a philosopher.
4. There is a book that is longer than Les Fleurs du Mal.
5. For any book written in French, there is a longer book.
6. It is not the case that there is a book that is longer than itself.
7. No book is written by a philosopher.
8. Every book is written by a mathematician.
9. It is not the case that every book is written by a philosopher.
10. For every book written in French and every philosopher, it is not the case that the book is written by the philosopher.
11. Every book written by a mathematician is a book written by a philosopher.
12. For any book written by Baudelaire, Begriffsschrift is a longer book written by G. Frege.
13. For any book written by Baudelaire, there is a longer book written by G. Frege.
14. There is someone who is both a philosopher and a mathematician and who has written a book in French.
15. Any book that is longer than Begriffsschrift or An Essay Concerning Human Understanding is longer than Les Fleurs du Mal.

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01:43

Problem 6

Take the blocks world described in the example in Section 4.2.1 and add to the domain one additional object, $p$, the floor. A block may now be on another block, or it may be on the floor, $p$. Thus, Oxy is now interpreted as: block $x$ is on $y$, where $y$ may be either a block or the floor. Suppose that the following are true in this blocks world:
1. There is a blue block on the floor with a yellow block on this blue one and a green block on this yellow one.
2. There is a blue block on the floor with a yellow block on this blue one and a blue one on this yellow one.
3. No yellow block is clear.
4. There is a clear blue block and a clear green block.
5. For any yellow block, there is a block on this yellow one.
6. Not all blue blocks are clear.
7. Any green block is clear.
8. All blocks on the floor are blue.
9. Any block that is on a yellow one is either green or blue.
Symbolize these sentences. Draw a sketch of the arrangement of the floor and the six blocks in this world. Show the colors of the blocks.

Mayukh Banik
Mayukh Banik
Numerade Educator

Problem 7

Let $B, G, Y, C, O, p$ have the same meanings as in the previous exercise. Symbolize the following sentences and construct a blocks world interpretation that satisfies them. Use as many blocks as you need, but try not to use more than are necessary. Name each of the blocks in your interpretation with an individual constant, say what colors they are, and describe their arrangements with respect to each other and the floor.
1. There is a green block on a green block.
2. For every blue block, there is a yellow one on this blue one.
3 . There is a clear yellow block on the floor.
4. There is a blue block on the floor with a yellow block on it.
5. There is a green block on a yellow one.
6. There is a green block on the floor.
7. There is a yellow block such that it is on a blue one and a blue one is on it.
8. Every block is on something. [Remark. The only individual objects in the domain are the blocks (of types $B, Y, G$ ) and the floor.]

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03:02

Problem 8

For each of the following sets of sentences, give a suitable interpretation to show that the set is satisfiable. Informally justify the truth values of the sentences under your interpretation.
1. $\{A b, B b, C b,(\exists x)(A x \wedge \neg(B x \vee C x)),(\forall x)(C x \rightarrow A x)\}$
2. $\{(\exists y)(B y \wedge C y), \neg(\forall z)(C z \rightarrow A z),(\forall x)((B x \wedge C x) \rightarrow A x)\}$
3. $\{((\exists x) A x \wedge(\exists x) B x), \neg(\exists x)(A x \wedge B x),(\forall x) \neg R x x$,
$$
\begin{aligned}
& (\forall x)(\forall y)(R x y \rightarrow R y x),(\forall x)(A x \rightarrow(\exists y)(B y \wedge R x y)), \\
& (\forall x)(B x \rightarrow(\exists y)(B y \wedge R x y))\}
\end{aligned}
$$
4. $\{(\exists x)(A x \wedge \neg B x),(\exists x)(B x \wedge \neg A x)$,
$$
\begin{aligned}
& (\forall x)(A x \rightarrow(\exists y)(B y \wedge R x y)), \\
& (\forall x)(B x \rightarrow(\exists y)(A y \wedge R x y)),(\exists x)(A x \wedge B x)\}
\end{aligned}
$$
5. $\{(\forall u)(\forall v)(\forall w)((R u v \wedge R v w) \rightarrow R u w)$,
$$
\neg(\exists x)(\exists y)(R x y \wedge R y x),(\forall x)(\exists u) R x u\}
$$
6. $\{(\forall u)(\forall v)(\forall w)((R u v \wedge R v w) \rightarrow R u w)$,
$$
\neg(\exists x)(\exists y)(R x y \wedge R y x)\}
$$
7. $\{(\forall x)(A x \rightarrow \neg B x),(\forall x)(A x \rightarrow(\exists y)(B y \wedge L x y))$,
$$
(\exists x)(A x \wedge \neg(\exists y)(A y \wedge L y x)),(\forall x)(\forall y)(L x y \rightarrow \neg L y x)\}
$$
8. | $(\forall x)(A x \rightarrow B x),(\forall x)(A x \rightarrow(\exists y)(C y \wedge R x y))$,
$$
\begin{aligned}
& (\exists x)(A x \wedge \neg B x),(\exists x)(B x \wedge(\forall y)(C y \rightarrow \neg R x y)), \\
& (\forall x)(\forall y)(R x y \rightarrow R y x) ?
\end{aligned}
$$

Amy Jiang
Amy Jiang
Numerade Educator
02:27

Problem 9

Use the general definition of an interpretation to show that the following are valid:
1. $(\forall x)(F x \vee \neg F x)$.
2. $(\forall x)((A x \wedge B x) \rightarrow B x)$.

Dharmendra Jain
Dharmendra Jain
Numerade Educator

Problem 10

Use the general definition of an interpretation to prove that
$$
\neg(\forall x) H x
$$
is equivalent to
$$
(\exists x) \neg H x .
$$
If $H$ means honest, we have:
It is not the case that everyone is honest.
is equivalent to
Someone is not honest.

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02:06

Problem 11

Use a counterexample interpretation to show that
$$
(\forall x)(A x \rightarrow(B x \vee \neg C x))
$$
is not valid.

Mohamed Mohamed
Mohamed Mohamed
Numerade Educator
02:57

Problem 12

Use a counterexample interpretation to show that
$$
(\forall y)(\forall z)((G y \wedge S y z) \rightarrow G z)
$$
is not valid.

Vysakh M
Vysakh M
Numerade Educator

Problem 13

Use a counterexample interpretation to show that $L a b \rightarrow(\exists x) L x x$ is not valid.

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01:04

Problem 14

Show that $(\exists x) P x \rightarrow P a$ is not valid, but that $(\exists x)((\exists x) P x \rightarrow P x)$ is valid.

Christopher Stanley
Christopher Stanley
Numerade Educator
05:32

Problem 15

Show that the following are tautologous:
1. $P a \rightarrow(P a \vee(\forall x) G x)$.
2. $(\neg(\forall x)(A x \wedge B x) \rightarrow(\forall x)(A x \wedge B x)) \rightarrow(\forall x)(A x \wedge B x)$.
3. $\neg((\forall x) R x x \rightarrow(\forall x)(\forall y)(R x y \rightarrow R y x)) \rightarrow(\forall x) R x x$.

Rosina Dapaah
Rosina Dapaah
Numerade Educator
03:02

Problem 16

Use counterexample interpretations to show that:
1. $(\forall x) A x$ is not a consequence of $(\exists y) A y$.
2. $G b$ is not a consequence of $(\exists x) G x$.
3. $(\exists x) \neg B x$ is not a consequence of $\{(\forall x)(A x \rightarrow B x),(\exists x) \neg A x\}$.
4. $R a b$ is not a consequence of $(\exists x) \operatorname{Rax}$.
5. $(\exists x) B x \rightarrow P$ is not a consequence of $B c \rightarrow P$.
6. $(\exists x)(\forall y) L x y$ is not a consequence of
$$
\{(\forall x)(\forall y)(\forall z)((L x y \wedge L y z) \rightarrow L x z)\} .
$$
7. $(\forall x) \neg B x$ is not a consequence of $\{(\forall x)(A x \rightarrow B x),(\exists x) A x\}$.
8. $(S \wedge(\exists x) B x)$ is not a consequence of
$$
\{(\forall x)(A x \rightarrow B x),(B c \rightarrow S),(\exists x) A x,(R \vee A c)\} .
$$
9. $(\exists x)(A x \wedge B x) \rightarrow(\forall x) B x$ is not a consequence of
$$
\{(\exists x) A x,(\forall x)(A x \rightarrow(\exists y)(B y \wedge R x y))\} .
$$

Amy Jiang
Amy Jiang
Numerade Educator
04:16

Problem 17

Sketch an $S C$ argument (i.e., cite the rule(s) you use in order of their use) to show that
$$
P a \rightarrow(\exists x) F x
$$
is a tautological consequence of
$$
\neg(P a \wedge \neg(\exists x) F x) .
$$

Clarissa Noh
Clarissa Noh
Numerade Educator
03:50

Problem 18

Sketch an $S C$ argument (i.e., cite the rule(s) you use in order of their use) to show that
$$
(\forall y) R y y
$$
is a tautological consequence of
$$
\{(\forall z) G z \rightarrow((\forall y) R y y \vee T a b),(\forall z) G z, \neg T a b\} .
$$

Clarissa Noh
Clarissa Noh
Numerade Educator

Problem 19

Consider the following: From $\Gamma=\{M a \rightarrow \neg(\exists y) F b y, M a\}$, we may infer
$$
\neg(\exists y) F b y,
$$
from which we may infer
$$
\phi=(\forall y) \neg F b y .
$$
Suppose that someone claims that this line of reasoning establishes that $\phi$ is a tautological consequence of $\Gamma$. Why is this claim mistaken?

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06:32

Problem 20

For each of the following arguments, give a $P C$ derivation of the conclusion from the premise(s). The conclusion is indicated by $/ \therefore$.
1.
$$
\begin{gathered}
(\forall x)(\forall y)(R x y \rightarrow \neg R y x) \\
/ \therefore(\forall x) \neg R x x
\end{gathered}
$$
2
$$
\begin{aligned}
& (\forall x)(\forall y) R x y \\
& \quad / \therefore(\forall x)(\forall y)(R x y \rightarrow R y x)
\end{aligned}
$$
3.
$$
\begin{gathered}
(\forall x)(\forall y)((D x \wedge M y) \rightarrow T x y) \\
\quad \therefore(\forall z)((D z \wedge M z) \rightarrow T z z)
\end{gathered}
$$
[Compare with: Suppose that, for any $x, y$, if $x$ is a demon and $y$ is a monster, then $x$ talks to $y$. Therefore: If anyone is both a demon and a monster, then he (or she) talks to himself.]
4.
$$
\begin{aligned}
& (\forall x)(\forall y)((G x \wedge N x y) \rightarrow L y) \\
& (\exists x)(G x \wedge N x a) \\
& \quad / \therefore(\exists z) L z
\end{aligned}
$$
5.
$$
\begin{aligned}
& (\forall x)((C x \wedge \neg(\exists y) P y x) \rightarrow O x) \\
& C a \\
& (\forall z) \neg P z a \\
& \quad \therefore \therefore a
\end{aligned}
$$
6.
$$
\begin{gathered}
(\forall x)(\exists y)(F x \wedge G y) \\
/ \therefore(\forall x) F x
\end{gathered}
$$
7.
$$
\begin{aligned}
& (\exists x)(\operatorname{Rax} \wedge(R x b \wedge R b a)) \\
& (\forall x)(\forall y)(\forall z)((R x y \wedge R y z) \rightarrow R x z) \\
& \quad \therefore(\exists u) R u u
\end{aligned}
$$
8.
$$
\begin{aligned}
& (\forall x)(F x \rightarrow \neg G x) \\
& (\forall x)(\exists y)(F x \wedge G y) \\
& \quad / \therefore(\forall x)(F x \wedge G x) .
\end{aligned}
$$
9.
$$
\begin{aligned}
& (\forall x)(\forall y)(((A x \wedge A y) \wedge \neg I x y) \rightarrow(R x y \vee R y x)) \\
& (\forall x)(B x \rightarrow A x) \\
& \quad / \therefore(\forall x)(\forall y)(((B x \wedge B y) \wedge \neg I x y) \rightarrow(R x y \vee R y x))
\end{aligned}
$$
10. $(\forall x)(P x \rightarrow(\exists y) R x y)$
/ $\therefore(\forall x)(\exists y)(P x \rightarrow R x y)$

Clarissa Noh
Clarissa Noh
Numerade Educator
05:18

Problem 21

Consider the argument: A lizard is a reptile. Therefore, a tail of a lizard is a tail of a reptile. This can be paraphrased as: All lizards are reptiles. Therefore, any tail of a lizard is a tail of a reptile. Let $L x$ represent ' $x$ is a lizard', $R x$ represent ' $x$ is a reptile', and T $x y$ represent ' $x$ is a tail of $y$ '. The sentence
$$
(\exists y)(L y \wedge T c y)
$$
means ' $c$ is a tail of a lizard'. Symbolize the argument and derive the conclusion from the premise.

Willis James
Willis James
Numerade Educator
02:20

Problem 22

Prove that $(\exists x)(P x \rightarrow(\forall x) P x)$ is a theorem of predicate calculus.

Mohamed Mohamed
Mohamed Mohamed
Numerade Educator
02:29

Problem 23

Exercise E2.45 defined a kind of relational system called a quasiorder, which is reflexive and transitive. In predicate calculus, we can write the assumptions, or axioms, about quasi-orders as follows:
$$
\begin{gathered}
(\forall x) Q x x \\
(\forall x)(\forall y)(\forall z)((Q x y \wedge Q y z) \rightarrow Q x z)
\end{gathered}
$$
E2.45 also added a definition of an indifference relation:
$$
(\forall x)(\forall y)(I x y \leftrightarrow(Q x y \wedge Q y x)) .
$$
Using these three sentences as premises, give three derivations to prove that $I$ is an equivalence relation, i.e., prove in predicate calculus that $I$ is reflexive, symmetric, and transitive. You will need to use predicate calculus definitions of these three terms. For instance, to state that $I$ is reflexive, one writes $(\forall x) I x x$. To avoid confusion, write three separate derivations. Notice which of the premises are actually needed in each of the derivations.

Amy Jiang
Amy Jiang
Numerade Educator

Problem 24

Either derive the conclusion from, or else show that it is not a consequence of, the premises of the following argument.
$$
\begin{aligned}
& A \\
& (B \vee A) \rightarrow P a \\
& Q b \vee \neg P a \\
& \quad / \therefore Q b \rightarrow(\exists x)(C x \wedge Q x) .
\end{aligned}
$$

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06:32

Problem 25

By deriving a contradiction, show that the following set of sentences is unsatisfiable:
$$
\begin{gathered}
\Gamma=\{(\forall x)(\forall y)(\forall z)(B x y z \leftrightarrow((O x y \wedge O z x) \vee(O x z \wedge O y x))), \\
(\exists x)(\exists y) B x y y, \quad(\forall x)(\forall y)(O x y \rightarrow \neg O y x)\} .
\end{gathered}
$$

Clarissa Noh
Clarissa Noh
Numerade Educator
06:11

Problem 26

Prove the Predicate Calculus theorems at the end of Section 4.3.2

Alex Roush
Alex Roush
Numerade Educator
06:11

Problem 27

Show that the converse of Predicate Calculus theorem number 9 (at the end of Section 4.3.2) is not valid.

Alex Roush
Alex Roush
Numerade Educator
01:04

Problem 28

Derive $(\exists x)((\exists x) P x \rightarrow P x)$ from $\varnothing$.

Christopher Stanley
Christopher Stanley
Numerade Educator
04:47

Problem 29

A person $x$ is a Great Joker (Jx) iff, for any person $y$ : $x$ makes a fool of $y$ iff $y$ does not make a fool of himself (or herself). Let Fxy denote ' $x$ makes a fool of $y$ '. Then we can define $x$ is a Great Joker by:
$$
(\forall x)(J x \leftrightarrow(\forall y)(F x y \leftrightarrow \neg F y y)) .
$$
From this definition, prove that Great Jokers do not exist, i.e., derive $\neg(\exists x) J x$. I suggest trying RAA.

Amy Jiang
Amy Jiang
Numerade Educator
05:55

Problem 30

Prove Metatheorem 4-16 about the quantifier exchanges.

Madi Sousa
Madi Sousa
Numerade Educator
01:47

Problem 31

Prove Metatheorem 4-17.

Manik Pulyani
Manik Pulyani
Numerade Educator

Problem 32

Prove the derivational counterpart of Metatheorem 4-17 by showing that the EG Rule is a derived rule. Use a proof schema (as in Metatheorem 4-18) that does not use EG to show that: if $\Gamma \vdash \phi[v / \kappa]$, then $\Gamma \vdash(\exists v) \phi$.

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01:57

Problem 33

Let $\Gamma$ be a set of $P C$ sentences, such that no sentence in $\Gamma$ has an occurrence of the constant $a$. Let $\sigma$ be a sentence with no occurrences of a. Without using Rule EA, give a direct semantical proof of the following:
$$
\begin{gathered}
\text { if } \Gamma \vDash F a \rightarrow \sigma, \\
\text { then } \Gamma \vDash(\exists x) F x \rightarrow \sigma .
\end{gathered}
$$

Nick Johnson
Nick Johnson
Numerade Educator

Problem 34

A corollary of the previous result is this: If $F a \rightarrow \sigma$ is a valid $P C$ sentence, then $(\exists x) F x \rightarrow \sigma$ is also valid. Yet Part 5 of E4.16 shows that $(\exists x) B x \rightarrow P$ is not a consequence of $B c \rightarrow P$. Discuss the differences between these two situations and explain why they are both true.

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03:38

Problem 35

Complete the proof of Metatheorem 4-24.

Geno Ellis
Geno Ellis
Numerade Educator
03:50

Problem 36

Construct a $P C$ derivation showing that
$$
(\forall x)(\forall y)(R x y \leftrightarrow L x y)
$$
implies
$$
(\forall x)(\forall y) R x y \leftrightarrow(\forall x)(\forall y) L x y .
$$
Notice how this result is related to the earlier discussion of (4.9) and (4.10). Also, construct a counterexample to show that the converse implication does not hold.

Clarissa Noh
Clarissa Noh
Numerade Educator

Problem 37

For any persons $x, y$ it is true that:
1. $x$ is an ancestor of $y$ if $x$ is the father of $y$.
2. $x$ is an ancestor of $y$ if $x$ is the mother of $y$.
3. $x$ is an ancestor of $y$ if (there exists $z$ such that $z$ is the father of $y$ and $x$ is an ancestor of $z$ ).
4. $x$ is an ancestor of $y$ if (there exists $z$ such that $z$ is the mother of $y$ and $x$ is an ancestor of $z$ ).
Notice that the last two conditions are recursive in a manner similar to that used before in the characterization of $H x y$. Let $A x y$ represent ' $x$ is an ancestor of $y$ ', $F x y$ represent ' $x$ is the father of $y$ ', and $M x y$ represent ' $x$ is the mother of $y$ '. Suppose that $a, b, \cdots, i$ are people, and we have the following data:
Fba, Mca, Fjb, Meb, Ffc, Mgc, Fhe, Mie .
Make $a$ the root node, and sketch this (family) tree. Now symbolize 1 through 4 in predicate calculus. Use these symbolized sentences as premises together with the family tree data. Give a predicate calculus derivation of Aia from all of these premises. The sketch of the family tree should help to guide you through the derivation.

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02:29

Problem 38

These are problems pertaining to the Omega Corporation, as described in Section 4.4.
1. Use the closures of (4.7) and (4.10), plus other relevant data in the knowledge base $\Omega$, to derive $\mathrm{Hma}$.
2. Let $S^3$ be a predicate that applies to real numbers such that, for any reals, $x, y, z$,
$$
S^3 x y z \text { iff } z=x+y .
$$
Suppose that $S^3$ is added to the set of predicates used to represent information about the Omega Corporation. Also, we add another predicate, $C$, such that $C x y$ holds iff $x$ is a department head and $y$ is the total annual compensation in dollars of $x$. We also add the closure of
$$
\left(H x \wedge\left(A x u \wedge\left(B x v \wedge S^3 u v t\right)\right)\right) \rightarrow C x t .
$$
Finally, assume that $S^3 45000500050000$ is an additional premise obtained as the result of a numerical computation. Now recall that $e$ is a member of the Omega Corporation. He (or she) receives both a salary and a bonus. Use the information in the text together with the preceding to prove that $\mathrm{Ce} 50000$. After this, please consider the next exercise.

Amy Jiang
Amy Jiang
Numerade Educator
07:43

Problem 39

Notice that the above formula for $C x t$ only applies to department heads. Also, the sum sentence, $S^3 45000500050000$, only applies to the particular person $e$. To compute the total compensation of the other two department heads, $g, j$, we need additional sum sentences. Also, suppose that we obtain additional information that the boss $a$ receives a salary of 100,000 dollars, and no bonus. Since there is exactly one boss, we can express this information by $A a 100000$. It would now be convenient to have the knowledge base imply the total annual compensation of any given person in the Omega Corporation. Add additional sentences to $\Omega$ so that this knowledge base has the following feature: Let $\kappa$ be any one of the constants denoting the people in OmegaCorp. Let $\tau$ be the number corresponding to the total annual compensation of $\kappa$. Then, for any such $\kappa$ and $\tau$, if $C \kappa \tau$ is a true statement of the total annual compensation of $\kappa$, then $\Omega \vdash C \kappa \tau$. Thus, from your expanded knowledge base $\Omega$, you should be able to derive $C a 100000, C e 50000, C h 25000$, etc. There are a number of ways of solving this problem; try to use no more sentences than are necessary.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
06:32

Problem 40

For each of the following sets of $P C I$ sentences, give a suitable interpretation to show that the set is satisfiable. Informally justify the truth values of the sentences under your interpretation. When interpreting a function symbol, be sure to specify a genuine function and show that it satisfies the conditions stated by the $P C I$ sentences.
1. $\{(\forall x)(A x \rightarrow x=a), \neg B a,(\exists x)(\exists y) x \neq y\}$
2. $\{(\exists z)(\forall x) x=z,(\exists x) A x,(\exists x) B x\}$
3. $1(\exists x)(\exists y)(\exists z)(x \neq y \wedge(x \neq z \wedge y \neq z)),(\forall x) f(x) \neq x$,
$$
(\forall x)(\forall y)(f(x)=f(y) \rightarrow x=y)\}
$$
4. $1(\exists x) A x,(\forall x)(A x \rightarrow \neg B x)$,
$$
(\forall x)(A x \rightarrow B f(x)),(\forall x)(B x \rightarrow A g(x))\}
$$
5. $\mid(\forall x)(\forall y)(f(x)=f(y) \rightarrow x=y)$,
$$
\begin{aligned}
& (\forall x) R x f(x), \quad(\forall)(\forall y)(R x y \rightarrow \neg R y x), \\
& (\exists x)(\forall y) x \neq f(y)\}
\end{aligned}
$$
6. $\{N a,(\forall x)(\forall y)((N y \wedge x=f(y)) \rightarrow N x)\}$
7. $\mid N a,(\forall x)(\forall y)((N y \wedge x=f(y)) \rightarrow N x)$,
$$
\begin{aligned}
& (\forall x) g(x, a)=x, \\
& (\forall x)(\forall y) g(x, f(y))=f(g(x, y))\}
\end{aligned}
$$

Clarissa Noh
Clarissa Noh
Numerade Educator

Problem 41

Give a counterexample interpretation to show that the conclusion is not a consequence of the premises. The conclusion is indicated by $1 \therefore$.
$$
\begin{aligned}
& (\exists x) A x \\
& (\forall x)(A x \rightarrow \neg B x) \\
& (\forall x)(A x \rightarrow B f(x)) \\
& (\forall x)(B x \rightarrow A g(x)) \\
& \quad / \therefore(\forall x)(A x \rightarrow g(f(x))=x) .
\end{aligned}
$$

Check back soon!
02:20

Problem 42

Give a counterexample interpretation to show that the conclusion is not a consequence of the premises. The conclusion is indicated by $/ \therefore$.
$$
\begin{aligned}
& (\forall x) p(x, a)=x \\
& (\forall x)(\forall y) p(x, s(y))=s(p(x, y)) \\
& \quad / \therefore(\forall y)(\forall z)(\exists x) p(x, y)=z
\end{aligned}
$$

Mohamed Mohamed
Mohamed Mohamed
Numerade Educator
01:49

Problem 43

Give a derivation of this $\mathcal{P C I}$ Theorem from $\varnothing$ :
$$
(\forall x)(\forall y)(x=y \rightarrow f(x)=f(y)) .
$$

Will Erickson
Will Erickson
Numerade Educator
01:53

Problem 44

Derive $(\forall x)((x=a \vee x=b) \rightarrow(f(x)=f(a) \vee f(x)=f(b)))$ from $\varnothing$.

Wendi Zhao
Wendi Zhao
Numerade Educator

Problem 45

For each of the following arguments, give a $P C I$ derivation of the conclusion from the premise(s). The conclusion is indicated by $/ \therefore$.
1.
$$
\begin{aligned}
& (\forall x) \neg R x x \\
& \quad / \therefore(\forall x)(\forall y)(R x y \rightarrow x \neq y)
\end{aligned}
$$
$$
\text { 2. } \begin{aligned}
&(\exists x) A x \\
&(\exists x) \neg A x \\
& / \therefore(\exists u)(\exists v) u \neq v
\end{aligned}
$$
3. $(\exists u)(\forall v)(A v \leftrightarrow v=u)$
$(\forall x) A x$
/ $\therefore(\forall x)(\forall y) x=y$
4. $(\forall x)(x=a \vee x=b)$
/ $\therefore(\forall x)(f(x)=f(a) \vee f(x)=f(b))$
5. $(\forall x)(\forall y)(g(y)=x \leftrightarrow y=f(x))$
$/ \therefore(\forall x) g(f(x))=x$
6. $(\forall x)(\forall y)(g(y)=x \leftrightarrow y=f(x))$
$/ \therefore(\forall x) f(g(x))=x$
7. $(\forall u)(\neg(\exists v) P v u \leftrightarrow u=o)$
$(\forall y)(\neg E y \rightarrow(\exists v) P v y)$
/ $\therefore$ Eo

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05:43

Problem 46

From $(\forall x)(\forall y)(R x y \rightarrow R y x)$ and $(\forall x)(\forall y)(R x y \rightarrow S f(x) f(y))$, derive $(\forall x)(\forall y)(S f(x) f(y) \rightarrow S f(y) f(x))$.

Dwijendra Rao
Dwijendra Rao
Numerade Educator
02:41

Problem 47

Derive the following from $\varnothing$ :
$$
(\exists x)(\exists y) f(x) \neq f(y) \rightarrow(\exists x)(\exists y) x \neq y .
$$

Lucas Finney
Lucas Finney
Numerade Educator
03:23

Problem 48

Let $\phi$ be the sentence $(\forall x)(\forall y)(\operatorname{Exy} \leftrightarrow f(x)=f(y))$. Using $\phi$ as the only premise, give $P C I$ derivations which establish that $E$ is an equivalence relation, i.e., prove that $E$ is reflexive, symmetric, and transitive.

Amany Waheeb
Amany Waheeb
Numerade Educator
04:53

Problem 49

The domain is the set of human beings. For any person $x$, let $m(x)$ denote the (biological) mother of $x$. Let $S x y$ represent ' $x$ and $y$ are sisters.' Let $B x y$ represent ' $x$ and $y$ are brothers.' Let $C x y$ represent ' $x$ and $y$ are matrilateral parallel cousins.' This means that $m(x)$ and $m(y)$ are sisters.
1. In $P C I$, write a formal definition of $C x y$ in terms of $S$ and $m$. This definition should have the form,
$$
(\forall x)(\forall y)(C x y \leftrightarrow \phi(x, y)),
$$
where $\phi(x, y)$ is a formula using $S$ and $m$.
2. Symbolize the sentence: For any $x, y$ : if $x$ and $y$ are brothers, then they have the same mother.
3. Symbolize: For any $x, y, z:$ if $x$ and $z$ are matrilateral parallel cousins, and $x$ and $y$ are brothers, then $y$ and $z$ are matrilateral parallel cousins.
4. Using the two $P C I$ sentences from 1 and 2 as premises, give a formal derivation of the $\mathcal{P C I}$ sentence obtained from 3 .

Clarissa Noh
Clarissa Noh
Numerade Educator
11:25

Problem 50

Prove statements (iii) - (vi) of Theorem 4-33. It may be helpful, though it is not necessary, to review the informal proofs of Theorem 2-48.

Vikash Ranjan
Vikash Ranjan
Numerade Educator