Determine whether the following are partitions of

A

=

Z

. Provide a detailed explanation/proof as to why or why not.

(a)

{

A

1

, A

2

}

, where

A

1

=

{

n

?

Z

|

n

= 2

k

for some

k

?

Z

}

and

A

2

=

{

n

?

Z

|

n

= 2

k

+ 1

for some

k

?

Z

}

.

(b)

{

A

1

, A

2

}

, where

A

1

=

{

prime numbers} and

A

2

=

{

composite numbers}

2. Determine whether the following are partitions of A = Z. Provide a detailed explanation/proof as to why or why not. (a) (41, An), where A, = {n E ZIn = 2k for some ke Z) and A, = {n ( Zyn = 2k + 1 for some ke Z). (b) {Al, Am), where An = {prime numbers) and Ay = {composite numbers)