Answer to Question #318022 in Discrete Mathematics for Donnel

Question

A. Provide formal of the validly of each of the following arguments.

1. I shall either play basketball or i shall go to the mall and watch the movie. therefore i shall either play basketball or go to a mall.
2. the Chocolate hills are not chocolate if I'm not the president of the Philippines. I'm not the president of the Philippines. therefore the Chocolate hills are chocolates.
3. if the summer is hot when we won't go on vocation in may. we'll either go on vacation inmay or we'll buy a boat (perhaps both). therefore, if the summer is hot, we'll buy a boat.
4. if Anna encore or write a program, she get an a. she's encoding and reading her lessons. therefore she'll get an A.

Answer 1

1. Let $A=$ I shall play basketball, $B=$ I shall go to the mall, and $C=$ I shall watch the movie.

Then $A\lor(B\land C)=$ I shall either play basketball or i shall go to the mall and watch the movie, and

$A\lor B=$ I shall either play basketball or go to a mall.

But $A\lor(B\land C) = (A\lor B)\land(A\lor C)$

If $A\lor(B\land C)$ is true, then both $A\lor B$ and $A\lor C$ are true.

So the argument is valid.

2. Let $A=$ I'm not the president of the Philippines, and $B=$ The chocolate hills are not chocolates.

Then $A\implies B$ is The Chocolate hills are not chocolates if I'm not the president of the Philippines, and $A\implies\lnot B$ is If I'm not the president of the Philippines, then the Chocolate hills are chocolates.

But the statements $A\implies B$ and $A\implies\lnot B$ are not equivalent. So the argument is not valid.

3. Let $A=$ The summer is hot, $B=$ We'll go on vacation in May, and $C=$ We'll buy a boat.

Then $A\implies\lnot B$ is If the summer is hot, then we won't go on vocation in May.

$B\lor C$ is We'll either go on vacation in May or We'll buy a boat.

$A\implies C$ is If the summer is hot, then we'll buy a boat.

Since $A\implies\lnot B$ is true and $B\lor C$ is true it follows that $A\implies C$ is true. So the argument is valid.

4. Let $A=$ Anna encodes, $B=$ Anna writes a program, and $C=$ Anna reads her lessons.

Then $(A\lor B)\implies C$ is true.

But $B\land C$ is true. So $B$ is true, and hence $B\implies C$ is true. So the argument is valid.