getting used to logical language

The first thing we need to do to thrive in csc165 is to try our best to get used to logical language. The first day I saw these mysterious symbols I felt that I'm done because I could barely accept this kind of language. Translating English to logical language is even harder. After 2 weeks' struggle I finally achieve some extent of progress.
At very beginning, Danny introduced a concept:
"for all p in D, f(p)" #sorry I don't know how to put logical symbols on the slogs.
f(p) is some kind of property that all p in D have in common.
It seemed pretty easy, however, I felt a little bit confused about the comma. What's the meaning?! I've been thinking: what if I change the comma to imply? Will that be equivalent? I went to Danny's office hour and csc165 help center and learned that the comma simply means "such that". The whole sentence is to state a property that all p in D have in common. I started to think that if I can use another way to understand it better: when we are stating a kind of property in this way, to some extent, it is also an implication! ( if p is in D, then f(p)). 
Speaking about implication, it is formed by an antecedent and a consequent. If the object meets the antecedent and  satisfies/ not satisfies the consequent, then we can tell the implication is true/false. However, it is really hard to believe that if the antecedent itself is wrong, then whatever the consequent is, the implication is always true! For this part it really took me so long to totally understand. This is, in fact, what called vacuous truth. I asked Danny and he said: if you want to falsify an implication, then you'll have to pick an object that meets the antecedent and dissatisfies the consequent. If the antecedent itself can never be met, then it means that you can never pick a counterexample to falsify the implication, and any implication that can not be falsify is, unexpectedly, true! Although this is really hard to understand, it does make sense to me and help me figure out this concept. From the progress of understanding vacuous truth, I also learned that it is equivalent to express "p implies q " as " not p or q", and to negate a implication we need to verify " p and not q". These concepts are actually saying the same thing.