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
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
The characteristic of sets {1, 4, 7 }, {2, 5, 8} and {3, 6, 9} is that difference between any two elements of these sets is a multiple of 3.
∴ (x,y) ∈ R1 ⇒ x – y is a multiple of 3
⇒ {x,y} ⊂ {1,4,7} or {x, y} ⊂ {2, 5, 8} or {x,y} ⊂ {3, 6, 9}
⇒ (x,y) ∈ R2.
Hence R1 ⊂ R2.
Similarly, { x, y} ∈ R2
⇒ {x, y}  ⊂ {1, 4, 7} or {x,y} ⊂ {2, 5, 8} or {x, y} ⊂ {3, 6, 9} ⇒ x – y is divisible by 3 ⇒ {x ,y} ∈ R1.
∴ R2 ⊂ R1.
Hence, R1Â = R2.
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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