No, it should be:
In this kind of context, an equal sign doesn't go between two statements. Rather, to express that two statements are equivalent, use the biconditional sign.
Or (and this is a picky technical point having not much substantive to do with the subject of images that these formulas are about), you could say:
" " stands for " ".
Anyway, all you need is:
That makes no sense. It is not well formed, not syntactically correct.
Probably what you mean is this:
x((x A & x B) -> x A B)
Or even just leave off the initial universal quantifier and let it be understood as tacit:
(x A & x B) -> x A B
I think you need to review basic symbolic logic to see how formulas are formed. I would suggest 'Logic: Techniques Of Formal Reasoning' by Kalish, Montague, and Mar.
It should be pointed out that is not a well-formed proposition (formula). The inductive definition of formulas goes like this. "... If P and Q are formulas, then is a formula... If P(x) is a formula that may contain a free variable x, then is a formula." So something that starts with is not a well-formed formula. (The culprit, of course, is not .)
If you write the statement as
reasoning about it would be easier.
No, you repeated a previous error.
(I'm tired of messing with LaTex tags, so I'm going to give you what you want in plain ASCII
~ stands for negation
e stands for membership
<-> will stands for material equivalence
u stands for binary union
U stands the universal quantification
v stands for disjunction
-> stands for material implication
not= stands for inequality)
~ y e f[AuB] <-> y not in f[AuB] <-> Ux((x e A v x e B) -> ~ y=f(x)) <-> UxeA UxeB y not= f(x) <-> Ux(x in AuB -> ~ y=f(x))
In other words:
it is not the case that y is in f[AuB]
if and only if
y is not in f[AuB]
if and only if
for all x, if (x in A or x in B) then ~ y=f(x)
if and only if
for all x in A and all x in B, ~ y=f(x)
if and only if
for all x in AuB, ~ y=f(x)
Take my suggestion to brush up on your symbolic logic?