MHB Valid Arguments: Sets Explained

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The discussion centers on logical arguments and set theory, specifically examining valid forms of reasoning like Modus Tollens and Transitivity. Participants clarify the correct interpretations of logical statements, emphasizing that the conclusion should be clearly indicated. The distinction between subset and element relations in set theory is also explained, highlighting that elements must appear distinctly within a set. Confusion arises regarding the notation used for conclusions, with clarification provided on the proper use of symbols. Overall, the conversation aims to enhance understanding of logical reasoning and set relationships.
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is this right ?6(1).
p$\implies$q,
$\lnot$q$\therefore\lnot$p,
Modus tollens

6(2).
p$\implies$q,
q$\implies$r,
q$\therefore$r,
Transitivity

6(3).
p$\implies$q,
q$\therefore$p.
Converse fallacy

7. I have no idea what I'm doing please explain.
 

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ertagon2 said:
is this right ?6(1).
p$\implies$q,
$\lnot$q$\therefore\lnot$p,
Modus tollens

6(2).
p$\implies$q,
q$\implies$r,
q$\therefore$r,
Transitivity

I think you mean $\therefore \; p\implies r$, right?

ertagon2 said:
6(3).
p$\implies$q,
q$\therefore$p.
Converse fallacy

This is all correct.

ertagon2 said:
7. I have no idea what I'm doing please explain.

The subset $\subset$ relation works like this: $A\subset B$ if and only if every element of $A$ is an element of $B$. So, for 7.A., look at every element of the set $\{1,2\}$, and see if they are in $A$ (they are). 7.B. is trickier. The relation "is an element of", $\in$, works differently from the $\subset$ relation. Sets can be elements of other sets, or they can be subsets of other sets. Neither implies the other. In this case, because $A=\{1,2,\{1,2\}\}$, and you have that nested $\{1,2\}$ sitting inside, then it follows that $\{1,2\}$ is an element of $A$. For something to be an element of another set, it's got to show up at the highest level, separated from other elements by commas. So, for example, if you define the set $C=\{1,2,\{3,4\}\}$, then the elements of $C$ are $1$, $2$, and $\{3,4\}$. Neither $3$ nor $4$ are elements of $C$. Does this help? Can you see why 7.C. is true and 7.D. is false?
 
Ackbach said:
I think you mean $\therefore \; p\implies r$, right?
This is all correct.
The subset $\subset$ relation works like this: $A\subset B$ if and only if every element of $A$ is an element of $B$. So, for 7.A., look at every element of the set $\{1,2\}$, and see if they are in $A$ (they are). 7.B. is trickier. The relation "is an element of", $\in$, works differently from the $\subset$ relation. Sets can be elements of other sets, or they can be subsets of other sets. Neither implies the other. In this case, because $A=\{1,2,\{1,2\}\}$, and you have that nested $\{1,2\}$ sitting inside, then it follows that $\{1,2\}$ is an element of $A$. For something to be an element of another set, it's got to show up at the highest level, separated from other elements by commas. So, for example, if you define the set $C=\{1,2,\{3,4\}\}$, then the elements of $C$ are $1$, $2$, and $\{3,4\}$. Neither $3$ nor $4$ are elements of $C$. Does this help? Can you see why 7.C. is true and 7.D. is false?

So I did answer everything correctly?
Can you please explain 6.(3)
Why is it $\therefore$p$\implies$r rather than p$\therefore$r <--this is what I meant
 
ertagon2 said:
So I did answer everything correctly?

Yes, I think so.

ertagon2 said:
Can you please explain 6.(3)
Why is it $\therefore$p$\implies$r rather than p$\therefore$r <--this is what I meant

The problem with writing $p\therefore r$ is that it is confusing. Someone looking at that might think you've concluded that $p$ is actually true, whereas it's only ever an assumption in the entire argument. Moreover, the word "therefore" or symbol $\therefore$ is meant to indicate that what follows is the conclusion. It's not usually meant to be used as part of the conclusion, or in the middle of it. Does that help?
 

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