Next: Monday, March 13, 2000 Up: Lecture Notes Previous: Wednesday, March 9, 2000

## Friday, March 10, 2000

```Starting Friday the class worked collectively on induction proofs.
A student led the class discussion (for class participation credit)
and a student wrote the class notes (again for participation credit
(homework 9)).

Proof of (Axyz.(x+y)+z = x+(y+z)) from student notes

Goal: (Axyz.(x+y)+z = x+(y+z))

| 1. a=a, b=b               arb. obj. intro. 2 times
| Goal: (a+b)+0 = a+(b+0)
| 2. (a+b)+0 = (a+b)+0      symmetry
| 3. (a+b)+0 = a+b          axiom 3, prop =, line 2, univ. elim.
| 4. (a+b)+0 = a+(b+0)      axiom 3, prop =, line 3, univ. elim.
| | 5. (a+b)+d = a+(b+d)    ind. hyp. [I use d for delta in the original --H.]
| | Goal:  (a+b)+d' = a+(b+d')
| | 6. [(a+b)+d]' = [a+(b+d)]'  prop =, line 5 (not ax 2!)
| | 7. (a+b)+d'   = [a+(b+d)]'  axiom 4, prop =, line 6
| | 8. (a+b)+d'   = a+(b+d)'    axiom 4, prop =, line 7
| | 9. (a+b)+d'   = a+(b+d')    axiom 4, prop =, line 8
| 10. (Az.(a+b)+z'   = a+(b+z'))  math induction, lines 4,5-9
11. (Axyz.(x+y)+z = x+(y+z))      multiple univ. intro. lines 1-10
```

Randall Holmes
2000-05-05