Short Answer TypeCheck the injectivity and surjectivity of the following functions:
f : N → N given by f(x) : x2Â
Check the injectivity and surjectivity of the following functions.
f : Z → Z given by f(x) = x2
Show that the Signum Function f : R → R, given by
           
is neither one-one nor onto
f : R → R, given by
  
It is seen that f(1) = f(2) = 1, but 1Â 
 f is not one-one
Now, as f(x) takes only 3 values (1, 0, - 1 ) for the element of - 2 in co-domain R, there does not exist any x in domain R such that f(x) = - 2
 f is not onto
Hence, the signum function is neight one-one nor onto.Â
In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.
f : R → R defined by f(x) = 3 – 4x
In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.
f : R → R defined by f(x) = 1 + x2
Consider the binary operation A on the set {1, 2, 3, 4, 5} defined by a ∧ b = min. {a, b}. Write the operation table of the operation ∧.
Let A = Q x Q. Let be a binary operation on A defined by (a, b) * (c, d) = (ac, ad + b).
Then
(i) Find the identify element of (A, *)
(ii) Find the invertible elements of (A, *)
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