Question

Prove:
\( J \cup K\) is denumerable where J is the set of all linear function with slope 1 and rational y-intercept, and K is the set of all linear function with slope 2 and and integer y-intercept

Answer 1

I came up with something but like I am not sure about it

Answer 2

There's only two things to prove here: that J union K is infinite and countable.

Since J is the set of all functions of the form y = x + r, where r is rational, J is infinite and countable since the set of rational numbers is infinite and countable.
Since K is the set of all functions of the form y = 2x + z, where z is an integer, K is also infinite and countablesince the integers are infinite and countable.
Since J and K are infinite and countable, their union is also infinite and countable.

Answer 3

yaaaaaaaaaa

Answer 4

THANNKKKSSSS :)