Real Analysis with Applications

[Dumpling]

Can anyone help me determine if the argument p, p ~ q, [(p ^ (~q)) -> (~p)] |- q is valid.
Given p, q and r be primitive statements.

Thank you

[kamil9876]

what do you mean by " p~q" ?

[Dumpling]

Oh my bad, I mean the question was suppose to be

p, ~q, [(p ^ (~q)) -> (~P)] |- q is valid.

So its not p ~ q its ~q.... sorry typo

[kamil9876]

Yeah it's valid simply because p, ~q, [(p ^ (~q)) -> (~p)] is a contradiction, as can be seen as follows:

Assume such a thing holds. Then p^(~q) is true since p and ~q are. Now that means that ~p is true. Which contradicts the fact that p is true. So such a thing cannot hold.

Who is teaching Real Analysis this semester?

[Dumpling]

Is it really that simple? And it is Richard Brak?

[Dumpling]

Also do you happen to know what => mean?

In this question [(p->q) ->r] => [(p^(~r)) -> (~q)]

I drew up a truth table and different truth values for (p->q)->r compared to (p^(~r)) -> (~q) .... so does it mean its not valid?

[kamil9876]

I think different people use different notation so I am not sure, check your notes or whatever and let me know if you can. I think I've once used "=>" instead of what you have been calling "->".  Whatever it is the following should be true:

[(p->q) ->r] => [(p^(~r)) -> (~q)] would mean that whenever [(p->q) ->r] holds then so does [(p^(~r)) -> (~q)], so the truth tables don't have to be the same, it just needs to be the case that in any row of the truth table, whenever [(p->q) ->r] holds then [(p^(~r)) -> (~q)] holds.

[Dumpling]

Oh thank you, in my notes it says => is "valid" and -> is the conditional. Well for => it says "a well defined logical meaning (and so is doing more than just separating the premises from the conclusion) ...

As well as for the first question do I need to draw up a truth table?

Thank you for helping me out

[Dumpling]

If P(A)=0.4, P(B)=0.5 and P(A or B')=0.1 -> Note: B' is B not occurring ~~

Find
(a). P(A and B)
(b). P(A or B)
(c). P(A' or B')
(d). P(A' or B)
(e). P(B|A)

My answers for question 1 were:
(a). 0.2
(b). 0.7
(c). 0.3
(d). 0.8
(e). 0.5


Question 2:

Let P(A)=0.3 and P(B)=0.6

(a). Find the probability of A or B when A and B are independent.
(b). Find P(A|B) when A and B are mutually exclusive.
(c). Find P(A|B') when A and B are mutually exclusive.

My answers for question 2 were;
(a). 0.72
(b). cannot exist (?)
(c). 0.5

Can anyone check those?

[kamil9876]

Hrmm so firstly Real Analysis includes propositional logic and now it involves probability? 

[Dumpling]

Just random stuff im helping out a little friend that posted.. my probability skills arent what they used to be

[Dumpling]

Help again with real analysis please
 
For this question prove the converse direction, that is, prove

p(x) => q(x) -> A C/ (subset symbol) B

where, for some non - empty set E,

A = { x element of E: p(x)}
B = { x element of E: q(x)}

Thank you