Question on De Morgan's Law

mzoodle

In one lesson, Jy says, the contapositive of "If Tom plays then Jerone and Simmi play too", is " If Jerone and Simmi do not play, then Tom won't play". However, it can be that J and S can play, even though T does not. If it would say "Only if Tom, then..", then ok I agree with this. But I do not understand why it cannot be that J and S can play, even though T does not?

bklsat05

Wouldn't it be If Jerone and or Simmi do not play, then Tom won't play

Contrapositive, deny the neccessary and you can't have the sufficient.

If Tom is playing then J & S have to be playing.
If J or S isn't playing then how can Tom play? If Tom is playing then J & S have to be playing but J or S isn't therefore Tom isn't

Does that make sense?

Only If introduces necessary so Only If Tom plays does J & S Play would be "J and S play only if Tom plays" would be:

J & S -> T
/T -> /J or /S

Basically, J and S playing requires Tom to have played/to play

VS

T -> J & S
/J or /S -> /T

Where Tom's playing requires J and S to have played/to play

If I am incorrect in my lawgic please let me know

akistotle

I think @bklsat05 has explained it perfectly

@mzoodle said:
In one lesson, Jy says, the contapositive of "If Tom plays then Jerone and Simmi play too", is " If Jerone and Simmi do not play, then Tom won't play". However, it can be that J and S can play, even though T does not. If it would say "Only if Tom, then..", then ok I agree with this. But I do not understand why it cannot be that J and S can play, even though T does not?

Does J.Y. say it cannot be "that J and S can play"? If sure he said "If Jerone and Simmi do not play". That does not mean J and S are not playing. (Hypothetical situation v fact).

I think you should review the lesson on The Contrapositive.
https://7sage.com/lesson/the-contrapositive/

Seeking Perfection

@mzoodle said:
In one lesson, Jy says, the contapositive of "If Tom plays then Jerone and Simmi play too", is " If Jerone and Simmi do not play, then Tom won't play". However, it can be that J and S can play, even though T does not. If it would say "Only if Tom, then..", then ok I agree with this. But I do not understand why it cannot be that J and S can play, even though T does not?

Let's strip out the words. The words are distractions.

Yout initial statement I'll work off is "If Tom plays then Jerone and Simmi play too."
T ---> J and S

The intuitive contrapositive is
/(J and S) to /T
This says If not (both J and S) then not T. In other words, if not J or not S then not T. I think this spot is where you got confused.
/J or /S ---> /T

Moving on...
"However, it can be that J and S can play, even though T does not."
This is right, but in no way contradicted by the previous.
T ---> J and S

T is the sufficient side of the conditional arrow. Knowing T is sufficient to know J and S. If T then J and S. If T plays J and S play too. If we don't have T then we could have both J and S or not. When the sufficient condition is negated the rule falls away.

J and S is the necessary side. J and S are needed to have T. If we don't have both J and S we cannot have T. Having J and S opens up the possibility that we have T. When the necessary is affirmed the rule drops away.

The above are all just different forms of the same thing. Eventually, you should be able to move between them without conscious thought.

"If it would say "Only if Tom, then..", then ok I agree with this. But I do not understand why it cannot be that J and S can play, even though T does not?"
At least by our starting statement, they can play even though he cannot.

"Only if Tom, then J and S."
This is a different relationship. Only if is slightly trickier.
As is it is a biconditional.
T <---> J and S

If we had "Only if Tom, can both J and S play." it is diagramed
J and S --->T
It is controposed
/T ---> /(Jand S)
/T ---> /J or /S

The first thing to do is learn how to do it with symbolic letters and not words, how to work in lawgic. Then learn how to translate to lawgic. Then use it. I hope this helps.

mzoodle

thanks,all!