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Problem: Prove that: In a classroom with n student, if there is a girl student, all students of this class are girl.

Solving: Let f(n) is the clause: In the class, if there is 1 girl student, all of n students are girl.

First: with n=1 ---> the clause f(1) is right (of course)

Second: with n=k, assume that the clause f(k) is right

we will prove that the clause f(k+1) is right.

Consider a set of (k+1) students {a1, a2, a3,....a(k),a(k+1)} (with a1 is the girl student)

  • with first k student A= {a1, a2, a3,....ak}: using the inductive supposition above, we have all student in A are girl.

  • Now we have B= {a1, a2, a3,....a(k-1),a(k+1)}: also using the inductive suppositon, we have have all student in B are girl.

So the f(k+1) clause is right.

Conclusion: In the class, if there is a girl student, all student are girl !!!!!!!!!!!

I can't find the wrong in this proof. Please help me. thanks

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Hint:

What happens to $A$ and $B$ for $k = 2$?

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  • $\begingroup$ exactly the proof breaks down at $k=2$, well-used induction example failure $\endgroup$ – Nikos M. Aug 16 '14 at 9:19
  • $\begingroup$ Oh. that's it... $\endgroup$ – T.Nhan Aug 16 '14 at 9:20
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In a proof by induction, going from the $k$ step to the $k+1$ step must work for every specific value of $k$. Does it in this case?

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