Answer to Question #260637 in Discrete Mathematics for BISHNU

Question

For this arguments, explain which rules of inference are used for each step. Same body in this class enjoys whole watching. Every person who enjoys whole watching cares about ocean pollution. Therefore, there is a person in this class who cares about ocean pollution.

Answer 1

(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.