List the elements

Question

Let X = {0,1,2}

1. C = {z:z = 2x and x element X}

2. C = {z:z = x + y where x element X and y element X}

3. C = {z:z element X and -z element X}

Answer 1

1. For what numbers are both the number and its double both in x? 

 

0 is in X, and 2*0 = 0 is also in X.  So 0 is one element of C

 

1 is in X and 2*1 = 2 is also in X.  So 1 is one element of C

 

2 is in X but 2*2 = 4 is NOT in X.  So 2 is not an element of C.

 

Since there are no other elements of X, that's all we will find.  So C = {0, 1}

 

2.  What numbers can be written as the sum of two numbers that are in X?  To figure this out, just try adding each element of X to another element of X and see what you get:

 

0 + 0 = 0

0 + 1 = 1

0 + 2 = 2

1 + 0 = 1

1 + 1 = 2

1 + 2 = 3

2 + 0 = 2

2 + 1 = 3

2 + 2 = 4

 

So the numbers that can be written as the sum of two elements of X are 0, 1, 2, 3, and 4.  So C = {0, 1, 2, 3, 4}

 

3. What numbers are in X and also have their negative in X?

 

0 is in X, and -0 = 0 is also in X.  So 0 is in C.

1 is in X, but -1 is NOT in X.  So 1 is not in C.

2 in in X, but -2 is NOT in X.  So 2 is not in C.

 

That's all the possibilities, since z has to be in X to start with.  So C = {0}