# Fallacious proof with induction [duplicate]

my teacher gave the following as an example of a fallacious proof:

We'll prove that a group of $n$ people are either all male or all female.

For $n = 1$ The claim says that a group containing 1 person is a group containing either only males or only females, which is correct. Inductive Step: Let's assume P(n) is correct. Let $A = \{{a_1,a_2,...a_n,a_{n+1}}\}$ be a group of $n+1$ people. We'll define $A' = \{{a_1,a_2,...a_n}\}$ and $A'' = \{{a_2,...a_n,a_{n+1}}\}$. There are two options, either $a_2$ is male or female. Let's assume $a_2$ is a male. Since A' and A'' are n-sized groups, they contain all females or all males, and since $a_2$ is a member of both, they both contain all males. But $A = A'\cup A''$, so A contains only males. A similiar approach is used if $a_2$ was a female.

What is the fallacy in this proof?