Comparing Mathematic Induction from Spivak & Courant

[dianzz]

Homework Statement

this is the 2 statement bout mathematica induction from different book ..fist is from spivak book : (1) P(1) is true
(2) whenever P(k) is true ,P(k+1) is true
and 2nd from courant book :
(1) if the assertion A_r is known to be true ,then the truth of assertion A_r+1 will follow
(2) the first statement A₁ is known to be true ..

Homework Equations

The Attempt at a Solution

what i think is the statement from spivak is more easy to digest and more reasonable in process , you must make sure that "one" is true first .. but courant say in diferrent way ..i don't know ..its look same in mathematica reasoning ??

 

[Mark44]

They both say the same thing, although the steps are in the opposite order. In Spivak, P(1), P(2), ..., P(k), P(k + 1), ... represent a sequence of statements. In Courant, the sequence of statements is written as A₁, A₂, ..., A_r, A_{r + 1}, ...

BTW, it's called mathematical induction. Mathematica is a computer program.

 

[HallsofIvy]

I have always thought of "induction" as "knocking over dominos. If you know
1) you can knock over the first domino and
2) anyone domino will knock over the next one

then you know that all dominos will fall.

But the answer to your question is exactly what Mark44 said originally: there is no difference except that they have stated the two conditions in reversed order.