This is me testing LATEX for my blog. Don’t judge me for this. :)
Question: If A1=cA2 and B1=cB2 prove that (A1→B1)=c(A2→B2).
Or in other words If the cardinality of A1 and B1 are equal to that of A2 and B2 respectively, then the cardinality of the set of functions from A1→B1 is equal to that of the set of functions from A2→B2
(from Notes on Set Theory, Yiannis Moschovakis, Chapter 2, Exercise 2.33)
Proof: Let σ:A1→A2 and ϕ:B1→B2 be two bijections.
And f:A1→B1 be any map. Define g=ϕ⋅f⋅σ−1. We claim that this map T:(f→g) is a bijection.
Is T well defined?#
Verify that T(f):=ϕ⋅f⋅σ−1 is indeed a map from (A2→B2)
Is T onto?#
Let g0 be any arbitary map from (A2→B2). Then T(ϕ−1⋅g0⋅σ)=ϕ⋅ϕ−1⋅g0⋅σ⋅σ−1=g0
Is T one-one?#
Let T(f1)=T(f2)
⟹ϕ⋅f1⋅σ−1=ϕ⋅f2⋅σ−1 ⟹ϕ−1ϕ⋅f1⋅σ−1σ = ϕ−1ϕ⋅f2⋅σ−1σ ⟹f1=f2
Note that since ϕ and σ are bijections, their inverses are well defined and function property of equal input gives equal output.
Hence T is a bijection from (A1→B1)→(A2→B2) and therefore (A1→B1)=c(A2→B2)