Answer to Question #190959 in Discrete Mathematics for Maaz

Question

For each of these arguments, explain which rules of inference are used for each

step.

a) “Doug, a student in this class, knows how to write programs in JAVA. Everyone who

knows how to write programs in JAVA can get a high-paying job. Therefore,

someone in this class can get a high-paying job.”

b) “Somebody in this class enjoys whale watching. Every person who enjoys whale

watching cares about ocean pollution. Therefore, there is a person in this class who

cares about ocean pollution.”

c) “Each of the 93 students in this class owns a personal computer. Everyone who

owns a personal computer can use a word processing program. Therefore, Zeke, a

student in this class, can use a word processing program.”

d) “Everyone in New Jersey lives within 50 miles of the ocean. Someone in New Jersey

has never seen the ocean. Therefore, someone who lives within 50 miles of the

ocean has never seen the ocean.”

Answer 1

The objective is to explain which rules of inference are used to obtain conclusions.

(a).

$Let\:J\left(x\right)=x\:knows\:JAVA,\:C\left(x\right)=x\:is\:in\:the\:class,$

$P\left(x\right)=x\:can\:get\:a\:high\:paying\:job,\:D=Doug$

Then the premises are: $\:C\left(D\right),\:J\left(D\right),\:\forall x\left(J\left(x\right)\rightarrow P\left(x\right)\right)$ and the conclusion is:

$\exists x\left(C\left(x\right)\wedge P\left(x\right)\right)$

The following steps can be used to establish the conclusion from the premises.

Step : Reason

1. $C\left(D\right)$ : Premise
2. $J\left(D\right)$ : Premise
3. $\:\:\:\forall x\left(J\left(x\right)\rightarrow P\left(x\right)\right)$ : Premise

Use the rule of inference universal instantiation $\:\forall xP\left(x\right)\Rightarrow P\left(c\right)\:for\:\forall \:x\left(J\left(x\right)\rightarrow P\left(x\right)\right)$

4. $J\left(x\right)\rightarrow \:P\left(x\right)$ : Universal instantiation

Apply Modulus ponens using steps 2 and 4 .

5.$\:P\left(D\right)$ : Modulus ponens from 2 and 4.

Apply conjunction using steps 1 and 5 and then apply existential generalization to the final step to get the conclusion.

6.$C\left(D\right)\wedge P\left(D\right)$ : Conjunction from 1 and 5.

7.$\exists x\left(C\left(x\right)\wedge P\left(x\right)\right)$ : Existential generalization of 6.

Therefore, the conclusion is that, there is someone in the class who can get a high paying job.

(b).

Let $P\left(x\right)=x\:$ cares about the ocean pollution, $C\left(x\right)=x$ is in the class, $\:W\left(x\right)=x$ enjoys whale watching.

Then the premises are $\forall x\left(W\left(x\right)\rightarrow P\left(x\right)\right),\:\exists x\left(W\left(x\right)\wedge C\left(x\right)\right)$ and the conclusion is

$\exists \:x\left(C\left(x\right)\wedge \:P\left(x\right)\right)$

The following steps can be used to establish the conclusion from the premises

Step : Reason

1. $\exists x\left(W\left(x\right)\wedge \:\:C\left(x\right)\right)$ : Premise
2. $\forall x\left(W\left(x\right)\rightarrow P\left(x\right)\right)\:$ : Premise

Use the rule of inference existential instantiation $\exists xP\left(x\right)\Rightarrow P\left(c\right)\:for\:\exists x\left(W\left(x\right)\wedge C\left(x\right)\right)$

3.$W\left(y\right)\wedge C\left(y\right)$ : Existential Instantiation

Apply simplification of 3.

4.$W\left(y\right)$ : Simplification of 3

Use the rule of inference universal instantiation $\forall xP\left(x\right)\Rightarrow P\left(c\right)\:for\:\forall x\left(W\left(x\right)\rightarrow P\left(x\right)\right)$

5.$W\left(y\right)\rightarrow P\left(y\right)$ : Universal instantiation of 2.

Apply modulus ponens using steps 4 and 5.

6.$P\left(y\right)$ : Modulus ponens from 4 and 5.

Apply simplification of 3.

7.$C\left(y\right)$ : Simplification of 3.

Apply conjunction using steps 6 and 7 and then apply existential generalization to the final step to get the conclusion.

8.$C\left(y\right)\wedge P\left(y\right)$ : Conjunction from 6 and 7.

9.$\exists x\left(C\left(x\right)\wedge P\left(x\right)\right)$ : Existential generalization of 8.

Therefore, the conclusion is that, there is a person in this class who cares about ocean pollution.

(C).

Let $C\left(x\right)=x$ is in the class, $\:\:P\left(x\right)=x$ owns a computer, $W\left(X\right)=x$ can use a word processing program, and $Z=zeke$.

Then the premises are$\forall x\left(C\left(x\right)\rightarrow P\left(x\right)\right),\:\forall x\left(P\left(x\right)\rightarrow W\left(x\right)\right),\:C\left(Z\right)$ and the conclusion is $\:\:W\left(z\right)$ .

The following steps can be used to establish the conclusion from the premises.

Step : Reason

1. $\:\forall x\left(C\left(x\right)\rightarrow P\left(x\right)\right)\:$ : Premise
2. $\:\forall \:x\left(P\left(x\right)\rightarrow \:W\left(x\right)\right)$ : Premise
3. $C(Z)$ : Premise

Use the rule of inference universal instantiation $\:\:\forall \:\:xP\left(x\right)\Rightarrow \:P\left(c\right)\:for\:\forall x\left(C\left(x\right)\rightarrow P\left(x\right)\right)$ .

4.$C\left(Z\right)\rightarrow P\left(Z\right)$ : Universal instantiation on 1

Apply modulus ponens using steps 3 and 4.

5.$P(Z)$ : Modulus ponens 3 and 4

Use the rule of inference universal instantiation $\forall xP\left(x\right)\Rightarrow \:P\left(c\right)\:for\:\forall x\left(P\left(x\right)\rightarrow W\left(x\right)\right)$

6.$P\left(Z\right)\rightarrow W\left(Z\right)$ : Universal instantiation on 2

Apply modulus ponens using steps 5 and 6.

7.$W\left(Z\right)$ : Modulus ponens 5 and 6

Therefore, the conclusion is Zeke is a student in this class and can use a word processing program.

(d).

Let lives$J\left(x\right)=x$ in NJ,$O\left(x\right)=x$ lives within 50 miles of the ocean,$\:\:S\left(x\right)=x$ has seen the Ocean.

Then the premises are:$\forall x\left(J\left(x\right)\rightarrow O\left(x\right)\right)$ and the conclusion is$\exists x\left(J\left(x\right)\wedge -S\left(x\right)\right)$.

The following steps can be used to establish the conclusion from the premises.

Step : Reason

1. $\forall x\left(J\left(x\right)\rightarrow O\left(x\right)\right)$ : Premise
2. $\exists x\left(J\left(x\right)\wedge -S\left(x\right)\right)$ : Premise

Use the rule of inference existential instantiation $\exists xP\left(x\right)\Rightarrow P\left(c\right)\:for\:\exists x\left(J\left(x\right)\wedge -S\left(x\right)\right)$

3.$J\left(y\right)\wedge -S\left(y\right)$ : Existential instantiation on 2

Apply simplification on step 3

4.$J\left(y\right)$ : Simplification on 3

Use the rule of inference universal instantiation $\forall xP\left(x\right)\Rightarrow P\left(c\right)\:for\:\forall x\left(J\left(x\right)\rightarrow O\left(x\right)\right)$

5.$J\left(y\right)\rightarrow O\left(y\right)$ : Universal instantiation on 1

Apply modulus ponens using steps 4 and 5

6.$O\left(y\right)$ : Modulus ponens from 4 and 5

Apply simplification on step 3

7.$-S\left(y\right)$ : Simplification on 3

Apply the conjunction using steps 6 and 7 and then apply existential generalization to the final step to get the conclusion.

8.$O\left(y\right)\wedge -S\left(y\right)$ : Conjunction from 6 and 7

9.$\exists x\left(O\left(x\right)\wedge -S\left(x\right)\right)$ : Existential generalization of 8

Therefore, the conclusion is someone who lives within 50 miles of the ocean has never seen the ocean.