Week 1 Math230
A) A) x
B) ¬∀x
C)
D) Some people did not take medication or placebo.
B) -∀x
C)
D) It is true some people did not take placebo or did not have migraines
B)
C)
D) It is true all patients did not have migraines or did not take placebo.
( can you please ignore the → above the x varibles they are their by mistake the current Ide enviorment im in is new to me, i didnt do it again haha. )
I will show the equivalence by verifying if each function on the sides of the = sign are equivalent to the other side of the = sign.
A) ok so i have one side as
we can distribute the - sign into our function for x for P and Q
we have
we can use our double negation rule for Q and turn it positive.
thus we have an equivalence to the other side.
our and symbol is flipped as we have done de morgans law.
these are equivalent expressions!
B) ok so we have one side as -∀x
We can begin by using de morgans law to see if these are recipicoral meaning they are equivalent when we inverse the properties i learned this in pre calculus. Which is why we use de morgans law.
we distribute the - sign from ∀ into our function for P and Q
we have (--
we use double negative to give us a positive
after solving with de morgans law we do not have
logical statements that are inverses of each other, fundamentally
these functions are not equivalent to each other!
C)
Is logically equivalent. I solved this mentally but I can share how to solve this.
ok so we use De morgans law to get the inverse and with this law we flip our expression to be the opposite of what they are.
For -∀ we distubute the negative sign into the pararntheis and we flip the ∀ to an ∃
we have almost the same expression on the other side
and when we use our double negative law they are the same however
we now have to flip our symbols as that is fundemental to de morgans law.
These are logically equivalent !
B) Is true as the predicate is
C) is not true
the predicate M suggest the x and y values got mail.
our expressions suggests some vaules of x got mail and all vaules of y got mail.
while its true some x s got mail, all the ys did not get mail.
this quantified statement is not true!