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Bijective function
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Let R be set of all real numbers, and
A = B = R*R
A function A> B is defined by
f(a,b) = (a+b,ab)
How to prove it is a bijective function?
discretemathematics
functions
asked
Nov 9, 2018
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Set Theory & Algebra
by
dan31
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You have to prove that sum and difference of 2 real numbers are always unique.
0
To prove a function is bijective, prove that it is oneone and onto i.e for every value of (a,b) there exists a unique value of f(a,b) and that for every value (x,y) present in the codomain, there exists a value (a,b) such that f(a,b) = (x,y).
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Let's there is a value $(x, y)$ in codomain of $RXR$. From there we can get $a=(x+y)/2$ & $b=(xy) /2$ and $a, b \epsilon R$. Thus proves Range = Co domain... This also proves for every $(x, y)$, there is unique pair or $(a, b) $. Thus proves function is bijective.
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Nov 9, 2018
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2019_Aspirant
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