True or False: If A = {x: x is the square of an Integer) and B = {z: z Is an Int

Question

True or False: If A = {x: x is the square of an Integer) and B = {z: z Is an Integer), then B is a subset of A. Let the function f from the integers to the natural numbers be defined by f(x) = x ast x (where ast is ordinary multiplication In the natural numbers). Determine whether f is one-to-one, and explain why it is or isn't. Let sets A and B be defined as follows: A = {al, a2) and B = {bl}. List as set(s) of ordered pairs all onto functions from A to B; If there are no onto functions from A to B, explain why not.

Explanation / Answer

2.

False

Square of an integer is positive

Hence all elements in A are positive integers. But B contains all lintegers

So, A is properly contained in B

So , B is not a subset of A

3.

It is one-to-one

Let, x, y so that:f(x)=f(y)

x*x=y*y

x^2-y^2=0

(x-y)(x+y)=0

x=y,x=-y

x=-y is not possible as we are dealing with natural numbers only.

Hence, x=y.

So f is one-to-one

4.

There is only one function from A to B

{(a1,b1),(a2,b1)}

which is onto as it maps to all elements of B which is b1 itself.

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