Tao Analysis I - A.1.3

Exercise A.1.3

Suppose that you have shown that whenever X is true, then Y is true, and whenever X is false, then Y is false. Have you now demonstrated that X and Y are logically equivalent? Explain.

The complexity of statements risks an error in trying to reason this out using only plain English, so let's go straight to truth tables.

The following table fills the truth values for the statement S, which are the conditions provided by the question.

X	Y	S	X<=>Y
F	F	T	T
F	T	F	F
T	F	F	F
T	T	T	T

Let's talk through how we filled in S:

• Whenever X is true, Y is true. This gives us row 4 as true. It also gives us row 3 as false because it contradicts the statement.
  
• Whenever X is false, Y is false. This gives us row 1 as true. It also gives us row 2 as false because it contradicts the statement.
  

Looking at the column S we see that it is only true when X and Y both have the same value. We recognise this as the logical equivalent, or the "if and only if" relation from the previous two exercises.

So, yes, we have demonstrated that X and Y are logically equivaelent. $\square$