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Question
Verify associativity for the following three mappings : f : N → Z0 (the set of non-zero integers), g : Z0 → Q and h : Q → R given by f(x) = 2x, g(x) = 1/x and h(x) = ex.
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Solution
Given that f : N → Z0 , g : Z0 → Q and h : Q → R .
gof : N → Q and hog : Z0 → R
⇒ h o (gof ) : N → R and (hog) o f: N → R
So, both have the same domains.
gof) (x)= g (f (x)) = g (2x) =`1/(2x)` ...(1)
(hog) (x) = h (g (x)) = h `(1/x) =e^(1/x)` ...(2)
Now,
( h o (gof)) (x) = h ((gof) (x)) = h `(1/(2x)) = e^(1/(2x))` [from (1)]
((hog) o f ) (x) = (hog) (f (x)) = (hog) (2x) = `e^(1/(2x)` [from (2)]
⇒ (h o(gof)) (x) = ((hog) o f) (x), ∀x ∈ N
So, h o (gof)= (hog) o f
Hence, the associative property has been verified
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