#Need help with logic!
river estuaryBOT
silk horizon
The first one is "p1 xor (p2 xor p3)" i think
pastel spindle
The first one is "p1 xor (p2 xor p3)" i think
Where does the “xor“ come from?
silk horizon
its just a binary operator similar to "and" and "or"
you can express it in terms of other binary operators in case you arent allowed to use it
p XOR q = ( p AND ~q ) OR (~p AND q)
pastel spindle
Ohhh… and how were you able to read it out of the truth function?
silk horizon
Intuition I guess. I just noticed that every row with 1 "1" evaluated to 1, every row with 2 "1"s evaluated to 0, the row with 3 ones evalueated to 1, and the row with 0 "1"s evaluated to 0. Which is kind of similar to the truth table for xor.
You also have to remember that every truth function can be expressed with binary operators. So if you just narrow your view to looking at 2 variables at a time it can help figure out which operators to use.
silk horizon
Not really sorry. Never seen that notation before.
fluid shell
Do you have an idea how to approach 2 and 3?😞
do you have a definition for $\vDash$?
elfin shadowBOT
Edward II
fluid shell
as for 1 there is a methodical way to do it btw but it almost always results in formulas much longer than you really need
pastel spindle
do you have a definition for $\vDash$?
It stands for truth
fluid shell
is that the whole statement you were given?
e.g. what does $\alpha \vDash\beta$ mean
elfin shadowBOT
Edward II
fluid shell
oh it's on the sheet
sterile inlet
Was studierst du?
fluid shell
so we know $\alpha\to\beta$ is true in any model, and we won't to show if we have a model where $\alpha$ is true, then also $\beta$ is true
elfin shadowBOT
Edward II
fluid shell
now if we have a model where $\alpha$ is true, we know $\alpha\to\beta$ is also true
elfin shadowBOT
Edward II
fluid shell
so what do we know $\beta$ has to satisfy for the latter to hold?
elfin shadowBOT
Edward II
pastel spindle
now if we have a model where $\alpha$ is true, we know $\alpha\to\beta$ is also ...
So the two tasks are intertwined?
fluid shell
you have to show that $\vDash\alpha\to\beta$ also means $\alpha\vDash\beta$, so you're going to have to use that $\vDash\alpha\to\beta$ at some point
elfin shadowBOT
Edward II
fluid shell
to show that $\alpha\vDash\beta$, you have to show that if we have a model that $\alpha$ fulfills, then so $\beta$ also fulfills that model
elfin shadowBOT
Edward II
fluid shell
in particular, as $\vDash\alpha\to\beta$ is fulfilled in any model, it is also fulfilled in a model that $\alpha$ fulfills
elfin shadowBOT
Edward II
pastel spindle
So do I have to make a truth table to show that or? 😭
fluid shell
no, these all follow from the definitions and the setup of what you want to prove
pastel spindle
I think that’s what my problem is, I don‘t know how to show those proofs
fluid shell
nothing I've said so far needs any proof:\\
We assume $\vDash\alpha\to\beta$. By definition, this means in any model $\alpha\to\beta$ is true ().\\
We want to show that in a model where $\alpha$ is true, so is $\beta$.\\
So we suppose we are in a model where $\alpha$ is true. As () applies to any model, $\alpha\to\beta$ is also true.\\ What truth values can $\beta$ take so that both of these can hold simultaneously? (to answer this, depending on how your course has set this up, you probably do need to use the truth table for $\to$)
elfin shadowBOT
Edward II
pastel spindle
I just saw, we didn’t have xor yet
Meaning I have to write a different formula for 1
fluid shell
Meaning I have to write a different formula for 1
you want a truth function that is true when
(p1 true and p2 false and p3 false) or (p1 false and p2 true and p3 false) or ...
pastel spindle
you want a truth function that is true when
(p1 true and p2 false and p3 false) ...
I wrote this for that