Show that the function f:N `rarr` N, given by f(n)=n-`(-1)^(n), AA n in N` is a bijection.

Show that the function f:N to N given by f(x) =2x , is one-one but not onto.

Show that the function f: N rarr N, given by f(1) f(2)= 1 and f(x) = x - 1 for every x gt 2, is onto but not one-one.

Show that f : N to N , given by f(x)= {(x+1", if x is odd"),(x-1", if x is even"):} is bijective (both one-one and onto).

Let f:N to N be defined by f(x) =x+2 . Then, find whether f is injective.

Let f : Z rarr Z be a function defined by f(n) =3n AA n in Z and g: Z rarrZ be defined by g(n) = {{:((n)/(3),"if n is a multiple of 3"),(""0, "if n is not a multiple of 3"):} Show that gof = I_(z) and fog ne I_(z) .

Let f : N rarr N be defined by f(n) = {{:((n+1)/(2), if "n is odd"),((n)/(2),"if n is even"):} Show that f is many one and onto function.

Show that the coefficient of a^m and a^n in expansion of (1+a)^(m+n) are equal.

2^n gt n , n is a natural number.

Show that the relation S defined on set N xx N by (a, b) S(c,d) rArr a+d=b+c is an equivalence relation.