Question

The sum of two rational numbers is rational. The first half of a proof for this statement is shown below.Suppose that x and y are both rational numbers. Since they are rational, we can write each one as a quotient of two integers. We will let = x=p1/q1 and y= p2/q2,where P1, P2, q1, q2 € Z and q1, q2 ≠0. Adding X and Y together, we get x+y = p1/q1 + p2/q2 Complete the proof that 2 + y is rational Be sure to justify each claim that you make,

Answer 1

`\begin{gathered} \text{For all }a,b\text{ in the set of integers} \\ a* b\text{ is in the set of integers} \end{gathered}`
`\text{ This means that the operation }*\text{ is closed on the set of integers}`

Also,

`\begin{gathered} \text{For all }x,y\text{ in the set of integers} \\ x+y\text{ is in the set of integers} \end{gathered}`
`\text{ This means that the operation }+\text{ is closed on the set of integers}`

We will now continue the proof and finish it.

From the image, we are at the point where we have

`x+y=(p_1q_2+p_2q_1)/(q_1q_2)`
`\begin{gathered} p_1,p_2,q_1,q_2\in Z \\ \text{ Then} \\ p_1q_2,p_2q_1\in Z \\ \end{gathered}`

Hence,

`p_1q_2+p_2q_1\in Z`
`\begin{gathered} \text{ Since} \\ q_1\\e0_{} \\ \text{and} \\ q_2\\e0 \\ \text{ then} \\ q_1q_2\\e0 \\ \text{and} \\ q_1q_2\in Z \end{gathered}`

Therefore, x + y is rational