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State with reason whether following functions have inverses: (i) \( f: \{ 1,2,3,4,\} \to \{10\}\) with \( f= \{(1,10),(2,10),(3,10), (4,10)\}\)

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  • To check if a function is invertible or not ,we see if the function is both one-one and onto.
  • A function $f: A \rightarrow B$ where for every $x1, x2 \in X, f(x1) = f(x2) \Rightarrow x1 = x2$ is called a one-one or injective function.
  • A function$ f : X \rightarrow Y$ is said to be onto or surjective, if every element of Y is the image of some element of X under f, i.e., for every $y \in Y$, there exists an element x in X such that $f(x) = y$.
Given $f:\{1,2,3,4\} \to \{10\}$ with $f=\{(1,10),(2,10,(3,10)(4,10)\}$
Step 1: Checking for one-one:
From the given definition of $f$, we can see that it is many-one, as for every ${1,2,3,4} \in f, f(1) = f(2) = f(3) = f(4) = 10$.
Therfore, the function does not have an inverse. We don't need to check further for onto.
answered Mar 19, 2013 by balaji.thirumalai

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