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Question: If A1=cA2A_1 =_c A_2 and B1=cB2B_1 =_c B_2 prove that (A1B1)=c(A2B2)(A_1 \to B_1 ) =_c (A_2 \to B_2 ). Or in other words If the cardinality of A1A_1 and B1B_1 are equal to that of A2A_2 and B2B_2 respectively, then the cardinality of the set of functions from A1B1A_1 \to B_1 is equal to that of the set of functions from A2B2A_2 \to B_2 (from Notes on Set Theory, Yiannis Moschovakis, Chapter 2, Exercise 2.33)

Proof: Let σ:A1A2\sigma : A_1 \to A_2 and ϕ:B1B2\phi : B_1 \to B_2 be two bijections. And f:A1B1f: A_1 \to B_1 be any map. Define g=ϕfσ1g = \phi \cdot f \cdot \sigma^{-1}. We claim that this map T:(fg)T:(f \to g) is a bijection.

Is TT well defined?

Verify that T(f):=ϕfσ1T(f):=\phi \cdot f \cdot \sigma^{-1} is indeed a map from (A2B2)(A_2 \to B_2)

Is TT onto?

Let g0g_0 be any arbitary map from (A2B2)(A_2 \to B_2). Then T(ϕ1g0σ)=ϕϕ1g0σσ1=g0T(\phi^{-1} \cdot g_0 \cdot \sigma) = \phi \cdot \phi^{-1} \cdot g_0 \cdot \sigma \cdot \sigma^{-1} = g_0

Is TT one-one?

Let T(f1)=T(f2)T(f_1) = T(f_2)

    ϕf1σ1=ϕf2σ1     ϕ1ϕf1σ1σ = ϕ1ϕf2σ1σ     f1=f2  \implies \quad \phi \cdot f_1 \cdot \sigma^{-1} = \phi \cdot f_2 \cdot \sigma^{-1} \ \implies \quad \phi^{-1} \phi \cdot f_1 \cdot \sigma^{-1} \sigma \ = \ \phi^{-1} \phi \cdot f_2 \cdot \sigma^{-1} \sigma \ \implies \quad f_1 = f_2 \

Note that since ϕ\phi and σ\sigma are bijections, their inverses are well defined and function property of equal input gives equal output.

Hence TT is a bijection from (A1B1)(A2B2)(A_1 \to B_1 ) \to (A_2 \to B_2 ) and therefore (A1B1)=c(A2B2)(A_1 \to B_1 ) =_c (A_2 \to B_2 )