I started out this chapter with new hope of learning new FOMy things, and got in return a section full of math hieroglyphics.
Hopefully this run through of what these symbols means will help us all out in using them in FOM (keyword hopefully). There’s actually a point to these symbols too, I asked Casey. Using these symbols instead of writing out statements in everyday language:
- Makes it impossible for your non-math major friends to read your homework
- Could be used as a really lame party trick if you want to try to convince your friends you know Egyptians hieroglyphics.
- Takes out any ambiguity that English statements provide, by defining each term in the statement.
The last one is the most important.
Okayyyy, here we go.
- Universal Quantifier: ∀: “for all”/”for every”, basically a conditional statement “in disguise”
- Existential Quantifier: ∃: “there exists”
- Free variable: a not bound variable. This (these) variable(s) are integral parts of the statement. This free variable depends on the quantified statement before it. For example, (∃ x ∈ ℝ) (x² = u). The variable u is free, because it depends on the definition of x, and is not quantified by ∀ or ∃. If a statement has at least one free variable, it is considered an open sentence.
- Bounded variable: also known as a dummy variable, or quantified variable. Basically a variable that is not integral to the explanation of a statement. For example, if you have (∀x∈ℕ)(x + 2 ≥ 2), you can explain the statement without using the term x. You could say “for any natural number, if you add two to that number, the answer will be greater than 2.” X is really just a place holder. Another way to think about it: x is just a placeholder in a definition, and x is quantified by the ∀. If there are only bounded variables in a statement, the whole statement becomes either true of false.
- Predicate: Basically the statement after your bounded variable is quantified. The predicate is the meat of the statement that connects the bounded variable and shows it’s relationship to the free variable. For example, (∃ x ∈ ℝ) (x² = u). The predicate is (x² = u).
- Open Sentence: For an open sentence p(x) and a set S, “(∀ x ∈ S) (p(x)) means that every element of S makes p(x) true. (∃ x ∈ S) (p(x)) means that there is an element in existence in the set S that makes p(x) true.
- Implication: the symbol : ⇒. Usually seen in between a hypothesis and a conclusion. P ⇒ Q.
- Negation: the symbol: ~. Basically turning a statement false. The negation of a conditional statement/universal statement is finding a counterexample. The negation of an existential statement is to find a contradicting true universal statement. For example, say I state that there’s a person somewhere in the library right now eating Frito’s loudly. The negation of this statement would be to say that for every person in the library right now, no one is eating Frito’s loudly (but that could still mean someone is quietly munching somewhere somehow). NOTICE, that the negation of an existential statement is a universal statement, and the negation of a universal statement is an existential statement. Coincidence? I think not.
Those are the highlights of this section that we went over in class as well. Hope they find you well!
Have a jammin’ weekend ya’ll.