Definitions

A variety of definitions will be listed below, without any text to label any of them. How many of them do you recognize?

  • \emptyset := \{ x ; x \neq x \}
  • A \cup B := \{ x \in U ; x \in A \lor x \in B \}
  • A \cap B := \{ x \in U ; x \in A \land x \in B \}
  • A^c := \{ x \in U ; x \notin A \}
  • A \subset B \leftrightarrow (x \in A \rightarrow x\in B)
  • \lnot(\lnot A) = A
  • \lnot (A \lor B) = \lnot A \land \lnot B
  • \lnot (A \land B) = \lnot A \lor \lnot B
  • (A \rightarrow B) \leftrightarrow (\neg B \rightarrow \neg A)
  • (x<y) \wedge (x=y) \wedge x>y
  • more
  • more
  • more
  • more

Appendix Z

The assignment was to make up some definition and then do some work with algebra, to show something true about it.

The students created the following definition:

a+* = (a+a)*(a+a)

  • 1+* = 4
  • 2+* = 16
  • 3+* = 36
  • 4+* = 64
  • 5+* = 100

How should we pla

The first comment was “Hey if we divide all these before we get squares!”

Well, since 4 = 1+*, we should show the formulas with “1+*”:

  • (1+*)/(1+*)=1
  • (2+*)/(1+*)=4
  • (3+*)/(1+*)=9
  • (4+*)/(1+*)=16

Hmm… does (x+*)/1+*)=x^2? Always?

We play with the definition and notice that it simplifies…

(a+a)*(a+a) = 2a*2a = 4a^2

(4a^2)/4 = a^2

Since 1+*=4, yes.