What is [Implications] ?

IMPLICATION basically is "If P then Q". P is called Antecedent or Assumption and Q is called Consequent or Conclusion. ( We can also use P=>Q to represent the relationship which is P implies Q).

When we say "If you are a Asian, then you are good at math", P represents "Asian" and Q represents "Good at math". However, it does not mean P causes Q. There are two types of implication we need to distinguish: Converse of implication and Contrapositive of implication. Converse version of the example is "If you are good at math, then you are a Asian." ( It does not have the same meaning with the original sentence) Contrapositive version of the example is "If you are not good at math, then you are not a Asian." (It means the same thing)

Then we summarize a list of "Everyday Language" for P => Q:
1. If P, (then) Q.
2. When(ever) P, (then) Q.
3. P is sufficient/enough for Q.
4. Can't have P without Q.
5. P requires Q.
6. For P to be true, Q must/need to be true/ is necessary.
7. P only if/ only when Q.
8. Not P unless/ if not Q.

When we want to falsify "P(x) => Q(x)" we just need to find an x such that P(x) is true but Q(x) is false.  If P(x) does not exist and Q(x) does not exist then it will be true.

In Equivalence, when we see "If P then Q, and if Q then P", it equals "P if and only if Q" = "P iff Q" using "<=>". Also, P implies Q, and conversely. P is true exactly when Q is true. P is necessary and sufficient for Q.

Important Note: (this is what I am confused about)

Every D that is a P is also a Q.

The common way to present it is :

less common way to present is :

Some D that is a P is also a Q.

The common way:

less common: 

The stuff we learn in CSC165 is getting confusing....BE CAREFUL!!!!!!