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