If f: N × N arrow N is defined by f((m,n)) = 2^{m-1}(2n-1), (m,n) N × N, then f is

Correct Answer is : Both one-one and onto

Correct Answer is : Both one-one and onto

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AP EAMCET Engineering 22 Sep 2020 Shift 2 Paper

Section: Mathematics

Question : 67 of 160

Marks: +1, -0

f: N \times N \rightarrow N

f((m,n))

2^{m-1}(2n-1)

\forall (m,n) \in N \times N

f

Solution:

The given function

f: N \times N \longrightarrow N

is defined by

f(m, n)=2^{m-1}(2 n-1), \forall(m, n) \in N \times N

Now, let

f((a, b))=f((c, d))

, where

a, b, c, d \in N

\Rightarrow 2^{a-1}(2 b-1)=2^{c-1}(2 d-1)

\Rightarrow 2^{a} \cdot b-2^{a-1}=2^{c} \cdot d-2^{c-1} \Rightarrow(a, b)=(c, d)

\therefore f

is a one-one function.

Now as

2^{m-1}

is a even number for

\forall m \in N-\{l\}

and

(2 n-1)

is a odd number for

\forall N \in N

, so the every natural number can be obtain by the for

2^{m-1} \cdot(2 n-1)

for some combination of

(m, n)

, so

f

is an onto function.

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