Help with Logic Questions | 65 Characters

[Plonker1]

Hi there, I desperately need your help :P

I was very ill for the past week and missed out on class time when we were taught the content. I attempted the homework questions but because I've had to breeze through the content I'm not sure if I have answered all the questions correctly or as well as I could have. Could you please review my four questions and inform me of the correct answers if I'm wrong?

Your help is VERY appreciated.

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View attachment 6457

My Answers:

1.
a) ¬a ∧ u ⇒ h
b) a ∧ u ⇔ h
c) a ⇒ ¬h ⊕ ¬a ⇒ h

2.
a) He does not speak or she does not jump
b) He speaks and she does not jump
c) Because he spoke she did not jump?

3.
a) I'm fine with this
b) Yes it is represented by p ⇒ q ⇒ r

4.
a) i. If all succeeds do not read the instructions. This is equivalent to p.
ii. Read the instructions only if all else fails
b) I'm fine with this

 

[Evgeny.Makarov]

b) a ∧ u ⇔ h

The sentence says, "If it's against the rules...". Therefore, the formula must have the form \(\displaystyle a\Rightarrow\ldots\). Second, "\(\displaystyle p\) only if \(\displaystyle q\)" means the converse of "\(\displaystyle p\) if \(\displaystyle q\)". The latter means \(\displaystyle q\Rightarrow p\) (the part after "if" is the premise and is therefore located to the left of \(\displaystyle \Rightarrow\)); so "\(\displaystyle p\) only if \(\displaystyle q\)" is \(\displaystyle p\Rightarrow q\). Taken together, "\(\displaystyle p\) if \(\displaystyle q\)" and "\(\displaystyle p\) only if \(\displaystyle q\)" form "\(\displaystyle p\) if and only if \(\displaystyle q\)", i.e., \(\displaystyle p\Leftrightarrow q\) (I am saying this simply for information; the equivalence is not used in (b)).

c) a ⇒ ¬h ⊕ ¬a ⇒ h

Here "but" is a synonym of "and". I am not sure how your course denotes conjunction, but usually it is denoted by \(\displaystyle \land\), \(\displaystyle \&\) or it is simply skipped like multiplication, as in \(\displaystyle pq\). The symbol \(\displaystyle \oplus\) usually denotes exclusive OR. Second, conjunction, just like multiplication, usually has the strongest precedence after negation, just like multiplication binds stronger than addition. So I would write the formula as \(\displaystyle (a\Rightarrow\neg h)\land(\neg a\Rightarrow h)\).

2.
a) He does not speak or she does not jump
b) He speaks and she does not jump
c) Because he spoke she did not jump?

I agree with a) and b). Concerning c), "because" is at best a complicated logical connective quite unlike others (AND, OR, etc.), and at worst it is not a connective at all. So I wouldn't worry about it. A good textbook should not include such question.

3.
a) I'm fine with this
b) Yes it is represented by p ⇒ q ⇒ r

The question asks: what does p ⇒ q ⇒ r mean: p ⇒ (q ⇒ r), (p ⇒ q) ⇒ r or something else? You can't answer p ⇒ q ⇒ r because that's what the question is about in the first place. I claim that the solution method means neither p ⇒ (q ⇒ r) nor (p ⇒ q) ⇒ r, but rather that $2x-6=0$ implies $2x=6$, and it in turn implies $x=3$.

4.
a) i. If all succeeds do not read the instructions. This is equivalent to p.

Please explain how you came up with this and why you think it is equivalent. Start by defining the inverse. Note that the negation of "all else fails" is "something else works".

ii. Read the instructions only if all else fails

I disagree. As I said,
\[
\text{"\(\displaystyle p\) only if \(\displaystyle q\)" is }p\Rightarrow q.\qquad(*)
\]
The statement $p$ has the form "all else fails $\Rightarrow$ you read the instructions". Now use the fact (*) to write it as a formula.

b) I'm fine with this

The problem does nort ask whether you are fine with the answer; it asks to justify it. Note that $p\Rightarrow q$ is equivalent to $\neg p\lor q$.

For the future, please read forum http://mathhelpboards.com/rules/.