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Question: let f z z be defined as fx...

Question details
Let
f : Z → Z
be defined as
f(x) =

x + 3 if x is odd
x − 7 if x is even
9. -7.69 points HunterDM3 2.3.024a My Let r:z-z be defined as x+3 x-7 if x is odd if x is even Show that f is a one-to-one correspondence To show f is a one-to-one correspondence we need to show that f is one-to-one and onto. First show that f is one-to-one. Notice that if f(x) is even, then x must be Selec-. Also, if f(x) is odd, then x must be Select. This is because in both cases of the function definition for f, f(x) differs from x by an Select-integer. An odd integer plus an odd integer is Selecwhile an odd integer plus an even integer is Selech. Now show that f(a)f(b) implies a b for both cases Suppose f(a) f(b) is odd. Then a and b areSelect. So which simplifies to a . Similarly if f(a) -f(b) is even then a and b are See. Substituting a and b into f we have the equation is one-to-one which simplifies to a-b. Therefore f To show that f is onto we need to show that for some arbitrary yeZ that there is an Xe Z such that f(x)y. Let yeZ be odd. Then y 7 is y +7) Similarly, suppose y eZ is even. Then y 3 isSelect Therefore f is both onto and one-to-one and has a one-to-one correspondence
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