Posted by: sclee09 | October 2, 2008

Implication

A few classes ago, we discussed the logic behind implications. Since it was a bit confusing to me in the beginning, I’ll use this blog post to try and explain the truth table for implications.

Implications come in the form “x implies y”, “if x then y”, and “y if x.” Essentially, these statements mean “if x is true, then y must be true.” Here are some examples:

1. If you go to Harvard, then you hate Yale.
2. I will lose weight if I don’t eat dessert.
3. x=3 implies x+2=5.

A helpful way to view an implication is as a promise. For example, Statement 1 can be rewritten as “You are guaranteed to hate Yale provided you go to Harvard.”

Now, lets break it down.

Case 1. Lets say you do go to Harvard (lucky you!). If you hate Yale, then the promise holds and Statement 1 is true. If you don’t hate Yale, then the promise is broken and Statement 1 is false.

Case 2. Now, you don’t go to Harvard. Since you don’t go to Harvard, you are not guaranteed to hate Yale. But, you are also not guaranteed to love (not-hate) Yale. So, regardless of whether you love or hate Yale, the statement “If you go to Harvard, then you hate Yale” is true.

In summary, a table:

 p q p–>q T T T T F F F T T F F T

Remember that a false statement implies anything.

Thanks to http://www.math.niu.edu/~richard/Math101/implies.pdf for a very clear explanation.

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Lastly, I wish it was still summer…