Implies what?

[LionelGoulet]

Would someone please explain to me the following sentence (taken from the book, Chapter 1):

The If-x-then-y function (also known as x -> y, or ‘‘x Implies y’’) returns 1 when x is
0 or when both x and y are 1.

I *never* understood "X Implies Y". I mean, "False Implies True"?

Can somebody break it down for me please?

[ybakos]

Not quite. That paragraph is directly related to the table in figure 1.2. In this case, the author is just stating that if x is 1, then the output is equal to whatever y is.

[LionelGoulet]

Sorry. Still doesn't make sense to me. Maybe it's the word "implies."
I can see that if X is 1, the output is Y, (thank you for showing that to me).
That *maybe* is a vague implication.
But if X is 0, the output is 1. The output has nothing to do with Y. Nothing is "implied" in that case.
So I'm pretty sure I don't "get" it.

[cadet1620]

"implies" in this case is shorthand for the Logical Implication operator (→) in Propositional Logic. (See http://en.wikipedia.org/wiki/Propositional_calculus.)

Given propositions p and q, p → q is defined to be True in all cases except True → False, which is False.

The confusing thing is that the result of the operator is about the validity of the statement "if p then q", not about the validity of q.

"If cows have 4 legs, then sheep have 6 legs." This is T → F = F. The statement is False because sheep don't have 6 legs.
"If cows have 6 legs, then ...." This is F → X¹ = T. Whatever comes after the "then" doesn't matter since cows don't have 6 legs. However worthless the statement may be, it is True.

I don't remember the formal usage of implication in Propositional Logic; that was nearly 40 years ago in school; imagine an engineer in an "Introduction to Logic" course taught by the Philosophy department!

--Mark

¹ X means "don't care" in equations and truth tables.

[LionelGoulet]

OMG! Thank you SO much! A conundrum of 40 years is resolved! The *statement* "If cows have six legs, then bananas are purple" is TRUE regardless of the fact that both parts are false! I get it!!!

Ooops! Spoke too soon ... http://www.youtube.com/watch?v=5h1FokjM-YQ