Problem 1.

Determine whether each of these functions from Z to Z is one-to-one:

a)f(n)=n-1

b)f(n)=n3

c)f(n)=n2 + 1

d)f(n)= n/2

Problem 2.

Determine whether f: ZZ  Z is onto if:

a)f(m,n) = 2m – n

b)f(m,n) = m2 – n2

c)f(m,n) = m+n+1

d)f(m,n) = m - n

Problem 3.

Give an example of a function from N to N that is:

a)one-to-one but not onto

b)onto but not one-to-one

c)both onto and one-to-one

d)neither onto nor one-to-one

Problem 4.

Let f:R  R and let f(x)  0 for all xR. Shor that f(x) is strictly decreasing if and only if the function g(x) = 1/f(x) is strictly increasing.

Problem 5.

Let f(x)= x2/3. Find f(S) if

a)S={-2,-1,0,1,2,3}

b)S= {0,1,2,3,4,5}

c)S={1,5,7,11}

Problem 6.

Suppose that g is a function from A to B and f is a function from B to C.

a)Show that if both f and g are one-to-one, then fg is also one-to-one.

b)If f and fg are one-to-one, does it follow that g is one-to-one?

Problem 7.

Let f be a function from the set A to the set B. Let S and T be subsets of

Show that:

a)f(S  T) = f(S)  f(T)

b)f(S  T)  f(S)  f(T).

Problem 8.

Draw the graph of the function:

a)f(x)= x + 1/2

b)f(x) =x-2 + x+2 /done/

c)f(x) = x-1/2 + 1/2

Problem 9.

Show that 3x = x + x+1/3 + x+2/3.

If x – integer, then it is true.

Let x = n + a where n – integer and 0  a  1/3.

Then x + x+1/3 + x+2/3 = 3n and 3(n+a) = 3n

Let x = n + a where n – integer and 1/3  a  2/3.

Then x + x+1/3 + x+2/3 = 3n + 1 and 3n+3a = 3n+1

Problem 10.

What is the value of each of these sums of terms of a geometric progression:

a){3 2j : 0  j  8} = 3 + 32 + 322 + … 328 = 3[1-29]/[1-2]

b){(-3)j : 0  j  8} = 1 + 1(-3) + 1 (-3)2 + …. 1 (-3)8 =

c){(3j -2j): 0  j  8} =

Solution: Geometric sequence a + aq + aq2+ aq3 + …. + aqn = a[1-qn+1]/[1-q]

Problem 11.

Determine whether each of these sets is countable or uncountable. For those that are countable, exhibit a one-to-one correspondence between the set of natural numbers and that set.

a)The even integers: 0, 2, -2, 4, -4,…………

b)The real numbers between 0 and ½ : not countable

c)Integers that are multiples of 7: 0, 7, -7, 14, -14, ……

Problem 12.

Show that subset of countable set A is countable.

If A countable then A = {a1, a2, a3, a4,…………..}

Lets take B  A, which means that only some elements in A are chosen.

We keep their order.