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Prove that for any cardinal numer $a,b,c$ that $a^b a^c = a^{b+c}$.

Start with three non-empty sets $A,B$ and $C$ such that $A∩B=\emptyset$,and show that $|A|^{|B|} \dot |A|^{|C|} = A^{|B| + |C|}$

Workings:

This would be done by establishing a bijection between $A^B \times A^C$ and $A^{B \cup C}$

But I'm not sure what the bijection could possible be.

Any help will be appreciated.

Note: I know of this Notation on proving injectivity of a function $f:A^{B\;\cup\; C}\to A^B\times A^C$.

But that uses notation and concepts I am not familiar with.

Martin Sleziak
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TillermansTea
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  • There is a natural candidate of a bijection: consider the map $(f,g)\mapsto f\cup g$. If $B$ and $C$ are disjoint, then $f\cup g$ is a function from $B\cup C$ to $A$ (why?) – Hanul Jeon Jan 24 '17 at 17:03
  • You may regard $f$ as a set of pairs, so the notation $f\cup g$ works and sometimes it defines a new function. – Hanul Jeon Jan 24 '17 at 17:04

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