I’m taking Keith Devlin’s course “Introduction to Mathematical Thinking” on Coursera for the next seven weeks. We’ve started with some thoughts and readings on set theory and logic.
And the first big thing in set theory is the way the stuff is written.
We invented shorthand to help us write stuff.
You may be old enough to remember Gregg shorthand. They discontinued the class the same year I intended to take it in high school. But I remember seeing it and really wanting to learn it.
The idea behind shorthand, both Gregg and any other, is that you have agreed upon symbols allow you to write things quickly, or more concisely.
Logic and set theory both use shorthand to do this. And in fact, you’re not a stranger to it either!
We live in an age of texting shorthand.
FYI and CC have been around for a long time. But LOL is relatively new. So is IMHO.
And every now and then I come across a brand-new one that completely throws me.
Logic and set theory shorthand works the same way. Consider the statement:
For every number that’s a positive number, we know that that number is bigger than the number -13.
That’s a math mouthful!
So instead we have invented some shorthand to make that easier.
We use variables like pronouns.
The first shorthand we use is variables. If the number were a guy, we would say,
If some number is positive, he’s bigger than the number -13.
Unfortunately numbers aren’t people. So instead of using he and she, we use x and y. So we say,
For every positive number x, we know that x is bigger than the number -13.
Then we get freaky!
Because math and set theory have been around a really long time, we have symbols that are easy to write with pencil and paper. But not so easy to write with typing!
Let’s start with one of my fav’s: $$\forall$$.
That upside down A stands for “for all” or “for every.” If we were to invent that notation now, we would probably use FA or something easy to text.
But we didn’t, so we’re stuck with it.
Here’s how we would use it in our example:
$$\forall$$ x such that x is a positive number, then x is bigger than -13.
You can also use it like this:
$$\forall$$ x such that x is a dish, you will wash x before you go to bed.
How about some more…
$$\exists$$ means “there exists” and can be used like this:
$$\exists$$ a dish in the sink, so you’re not going to bed yet.
means “therefore” or “thus” or “because of that, this will happen…” Here’s a way to use it:
$$\exists$$ a dish in the sink, you’re not going to bed yet.
$$\land$$ means “and” – to be used like this:
$$\exists$$ a dish $$\land$$ a glass in the sink, you’re not going to bed and you just might lose your phone privileges tomorrow.
Math isn’t a new language – it’s a shorthand of normal language.
And it isn’t really hard. Give this sentence a shot. I’ll be you can figure out what the symbols mean!
$$\exists$$ a dish $$\lor$$ a glass that isn’t washed $$\Rightarrow$$ you’re getting into serious trouble!
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