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Suppose that a, b, and y are all functions.
If ay=by and y is one-to-one and onto, prove that a=b.
This looked very straightforward at first then I realised that a and b are functions, not values. I am therefore stuck on what to do.
Any help is much appreciated.
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It is easy. Suppose that D is the domain of y and R is its range, since y is one to one then ∀ m ∋ D there is one and only one image in R and since y is outo ∀ n ∋ R there is a distinct inverse image "x" in D hence:
(a∘y)∘yˉ¹=(b∘y)∘yˉ¹ ⇒ a∘(y∘yˉ¹)=b∘(y∘yˉ¹) ⇒ a∘x= b∘x ⇒ a=b ( since x is a variable)
Note: the "ay" and "by" is a composition "∘" operation not multiplication "×" since if it was so, it could be solved by division as ordinary numbers.
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