Let f : X → Y be an invertible function. Show that the inverse of f-1 is f, i.e., (f-1)-1 = f.
Let f : X → Y be an invertible function. Show that the inverse of f-1 is f, i.e., (f-1)-1 = f.
Solution and Explanation
Let f : X → Y be an invertible function.
Then, there exists a function g: Y → X such that gof = IX and fog = IY
Here, f-1 = g.
Now, gof = IX and fog = IY
⇒ f-1 of = IX and fof -1= IY
Hence, f-1 : Y → X is invertible and f is the inverse of f-1
i.e., (f-1)-1 = f.
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Determine whether each of the following relations are reflexive, symmetric, and transitive.
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- Relation R in the set of N natural numbers defined as R={(x,y): y=x+5 and x<4}.
- Relation R in the set A={1,2,3,4,5,6} defined as R={(x,y): y is divisible by x}.
- Relation R in the set of Z integers defined as R={(x,y): x-y is an integer}
- Relation R in the set of human beings in a town at a particular time given by
- R={(x,y): x and y work at same place}
- R={(x,y): x and y live in the same locality}
- R={(x,y): x is exactly 7 am taller that y}
- R={(x,y): x is wife of y}
- R={(x,y):x is father of y}
Show that the relation R in the set R of real numbers, defined as
R = {(a, b): a ≤ b2 } is neither reflexive nor symmetric nor transitive.Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as
R = {(a, b): b = a + 1} is reflexive, symmetric or transitive.Show that the relation R in R defined as R = {(a, b): a ≤ b}, is reflexive and transitive
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