"A dolphin cannot fly and a bird can swim" is false.

Let variables P and Q stand for the 'positive' idea of verbs: 
(That is, we use 'can fly' rather than 'cannot fly'):

Let P = "A dolphin can fly."  
Let Q = "A bird can swim. 

Then only the first part in quotation marks, that is, 

"A dolphin cannot fly and a bird can swim"

(without "is false") in symbols is

~P & Q

Then annexing of "is false" means that we are negating that and we must put
parentheses around it and a ~ (negation symbol) before it.

~(~P & Q)

Demorgan's law says this is equivalent to

~~P v ~Q

The double negation before P makes it

P v ~Q

A dolphin can fly or a bird cannot swim.

That is one possible solution.  

However there is another equivalent statement we could get.

By commutation we get:

~Q v P

which in words is:

"A bird cannot swim or a dolphin can fly

Then we can use 'material implication' to get

Q -> P

which means

If Q then P 

which in words is

If a bird can swim, then a dolphin can fly.

That is another possible solution.

[All statements above which are in red are equivalent to the given statement.]

Edwin