Answer:
(i)
f : N N given by
f(x) = x2
Injectivity : Let x1,
x2
such
that f(x1) = f(x2)
is one ? one.
Surjectivity : Let y = 5
be any element
for y = 5
there is no elemnt in N
Hence f : N
N is not
onto.
(ii) f : Z
Z given by
f(x) = x2
Injectivity : Let x1
= 2 and x2 = ? 2 be two element to Z.
f(x1)
= (2)2 = 4; f(x2) = (?2)2 = 4
f (x1)
= f(x2) but x1
x2
f : Z
Z is not one
one.
Subjectivity : Let y = 5
Z be any
element
f(x) = 5
x2
= 5
x =
for y = 5
there is no element in Z
Hence
is not onto.
(iii) f :
given by f(x)
= x2
Injectivity : Let
x1 = 2 and x2 = ?2 be two element of R.
f(x1)
=(2)2 = 4; f(x2) = (?2)2 = 4
is not
one-one.
Surjectivity : Let
y = ? 5
be any
element
y = ?5, there
is no element in R.
Hence f :
is not onto.
(iv) f : N
N given by
f(x) = x3
Injectivity : Let
x1, x2
such
that
f(x1)
= f(x2)
is one-one.
Surjectivity : Let
y = 4
N be any
element
Hence f :
given by f(x)
= x3
Injectivity :Let x1,
x2
such
that
f(x1)
= f(x2)
is one-one.
Surjectivity : Let
be any
element
is not onto.
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