Math Week 2

In an unexpected and thrilling departure from the syllabus and our expectations, this week Math Proofs, our exciting new math course, covered Boolean logic.

Statements

In math, especially abstract math, it’s fun to categorize things that one would usually take for granted. For example, the previous sentence is an example of a statement. A statement is a thing that is either true or false. In logic, you can combine statements to form compound statements the truth value of which depends on the constituent statements. I.E. It is raining and I am in France depends on It is raining and I am in France.

Here are the ways we talked about to combine statements:

And and or seem pretty straight forward. Not is also simple (It makes I am in France into I am _not_ in France). But let’s use the idea of and, or, and not to demonstrate how truth tables work: (T is for True and F is for False)


P | Q | P and Q | P or Q | not P ——————————— T | T | T | T | F T | F | F | T | F F | T | F | T | T F | F | F | F | T —-

The idea here is pretty simple. The table shows all the possible values of P and Q, which are statements, along with the result of using and, or, and not. If you don’t quite get it, go through the table and assure yourself that everything in the P and Q column matches your intuition of what P and Q should be.

This gives us a good way to demonstrate how implication and the bi-conditional work:


P | Q | P -> Q | P Q ———————— T | T | T | T T | F | F | F F | T | T | F F | F | T | T —-

Implication is true, except when P is true and Q is false. This is because either P and Q are both true, and so P implies Q makes sense, or P is false, and so no matter what Q is, the problem is with P, and not your logic. My professor says this can be a bit difficult to grasp, so you might want to meditate on implication for a moment.

The bi-conditional is even easier. Either both P and Q are true, or they’re both false. Their fates are bound together.

Open Sentences

There’s only one thing left that we have yet to cover from last week, which is open sentences. An open sentence is like a statement with a variable, with a few rules. You have to say what that variable can be, and if you put the variable in the sentence, it must either be true or false.

Example!


P(f) : I am in f. where f is in {France, Belgium, Taiwan, Trouble}. —-

P(f) is an open sentence, because if you substitute any of those things in place of f, you have a statement. That’s all we really covered with open sentences, but I’m told they’re important, so hold onto that knowledge.

I’ll be back again next week, folks, but I’m not sure where the syllabus will take us.