Implications or conditional statements are molecular statements of the form P -> Q

• P is called hypothesis
• Q is called conclusion

P -> Q is false only when P is true and Q is false

For example, Pythagoras theorem can be written as an implication statement. We can state that, If a, b and c are the base, height and hypotenuse of a right angle triangle, then

a^2 + b^2 = c^2

So if a triangle is a 'right angled triangle', then it implies that this formula is correct and valid.

Some initial considerations:

• When hypothesis is false, we make no conclusions
• The whole implication statement is true or false depending upon the conclusion

P	Q	P -> Q
false	false	true	← Why?
false	true	true	← Why?
true	false	false	← This will only be false
true	true	true	← Makes sense :)

Why P -> Q is true when P is false?

This confuses sometimes, to get an intuition of this.

Let's try to understand this:

This is clear that:

• P = true and Q = false, then P -> Q = false
• and, P = true and Q = true, then P -> Q = true
• but if, P = false and Q either false or true, then by default we consider P -> Q = true

We can understand this by taking the Sets approach,

Let us assume P and Q as two Sets, implication says that P -> Q, where -> is implies.

So, P implies Q, that means P is the subset of Q.

P will be inside Q.

So, a given statement can be represented like the above diagram.

Let's consider a point x as the circumstance for the statement, i.e., it will determine the value to be true or false for P and Q and eventually for the implication.

So, depending upon the values for P and Q, we can place the point x on the venn diagram.

Suppose, P is true and Q is true, then

This means that x is inside both P and Q,

• x is inside P = true
• x is inside Q = true

As we can see in the above diagram, this is a possible circumstance, as whatever is in P will automatically in Q. So P -> Q = true

Suppose, P = true and Q = false, this is not possible as we already saw that whatever is in P will automatically in Q. So, P -> Q = false

Now suppose, P = false and Q = true / false (anything). Then, there are two possible placement of point x in venn diagram:

Q = true

Q = false

Looking at both of the cases, we can say that both of them are possible. So, P -> Q = true, when P is false and Q is either true or false.