Of all Mathematics I learned at my undergraduate courses I value most the simple but powerful two logics which I consider are very useful in understanding the world.

They are,

1. x ⇒ y (*x implies y* or *if x then y*)

2. x ⇔ y ( *x if and only if y*)

These two logics apply in most of the things we see/hear in our every day life. To understand them one need not have great learning on Mathematics or Logic. It is very intuitive, yet very subtle in their application. Some times it may seem too simple. But when you think carefully you understand that you need to be very careful in applying them with facts.

First simply mean that if x is true then y must be true. It is that simple. The only fact we know is that if x is true then y must be true. Take x and y to be any two facts that you face in life. For example let x be the fact that you buying product O from company M and y be the fact that you get discount in buying O.

So here our **expressed** logic is

If you buy O from M then you will get discount.

This could be any marketing propaganda that you are targeted at in your every day life. So what do you think?. What if you don’t buy O?. Then you won’t get discount?. The answer is, If you don’t buy from M, there is no more implication about your discounts with regard to buying from M. Of course you may still get discount by buying from some other company. May be not. We cannot come to any precise conclusion about your discounts from the given logic.

x ⇒ y is not equalent to (not x) ⇒ (not y). But sadly this is something that most people take without giving proper heed to the facts.

But here is an equivalent logic

x ⇒ y is equivalent to (not y) ⇒ (not x)

So under the context of our example we can say that

If you are not getting discount in buying your O then you haven’t bought O from M. Yes this is true.

So what if x ⇒ y and (not x) ⇒ (not y) are both given as true logic?

Look carefully. (not x) ⇒ (not y) is equivalent to

(not (not y)) ⇒ (not (not x)). What does that mean?. It is simply y ⇒ x.

So we have both x ⇒ y and y ⇒ x. In other words x if and only if y. This logic is defined with the symbol x ⇔ y.

So when this apply to our example it says

If you buy O from M then you get discount and if you get discount then you have bought it from M. So in this case it is to be understood that only M company gives discount for O and no other company in this world give a discount for O. If some company express such logic it should be taken as very strong claim. You get discount for product O if and only if you get O product from our company.

When reading anything, hearing anything I tend to think precisely along these logics. In that way I believe I get no more than it really mean, and no less than it really mean. This result in very little chance of getting cheated at least.

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