Solution to Determine whether each of these functions from {a,b,c,d} to itself is one-to-one. a) f(a)=b, f(b)=a, … - Sikademy
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Archangel Macsika

Determine whether each of these functions from {a,b,c,d} to itself is one-to-one. a) f(a)=b, f(b)=a, f(c)=c, f(d)=d

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Let us determine whether the function from \{a,b,c,d\} to itself is one-to-one.

a) f(a)=b, f(b)=a, f(c)=c, f(d)=d.


Taking into account that preimage of each point is singleton: f^{-1}(a)=\{b\},\ f^{-1}(b)=\{a\},\ f^{-1}(c)=\{c\},\ f^{-1}(d)=\{d\},

we conclude that x\ne y implies f(x)\ne f(y) for any x,y\in\{a,b,c,d\},

and hence the function f is one-to-one.

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