- #1
ephedyn
- 170
- 1
Homework Statement
"Prove that the sum of two rational numbers is a rational number."
I just started on proof writing, so I'll just like to verify if I'm not missing anything here, and get some comments about the style.
The attempt at a solution
Theorem. If [tex]a,b \in \mathbb{Q}[/tex] then [tex]a+b \in \mathbb{Q}[/tex]
Proof. Given [tex]a,b \in \mathbb{Q}[/tex]
we have some [tex]p_1,p_2,q_1,q_2 \in \mathbb{Z}[/tex] satisfying
[tex]a=\dfrac{p_1}{q_1} \, b=\dfrac{p_2}{q_2}[/tex]
which implies that
[tex]a+b
=\dfrac{p_1}{q_1}+\dfrac{p_2}{q_2}
=\dfrac{p_{1}q_{2}+p_{2}q_{1}}{q_{1}q_{2}}[/tex]
By closure of [tex]\mathbb{Z}[/tex] under addition and multiplication, it follows from the assumption that [tex]p_1,p_2,q_1,q_2 \in \mathbb{Z}[/tex] that
[tex]p_{1}q_{2}+p_{2}q_{1}[/tex] and [tex]q_{1}q_{2}[/tex] are also integers.
Hence, [tex]a+b[/tex] can be expressed as the ratio of 2 integers. By definition, [tex]a+b[/tex] is rational. [tex]\square[/tex]
"Prove that the sum of two rational numbers is a rational number."
I just started on proof writing, so I'll just like to verify if I'm not missing anything here, and get some comments about the style.
The attempt at a solution
Theorem. If [tex]a,b \in \mathbb{Q}[/tex] then [tex]a+b \in \mathbb{Q}[/tex]
Proof. Given [tex]a,b \in \mathbb{Q}[/tex]
we have some [tex]p_1,p_2,q_1,q_2 \in \mathbb{Z}[/tex] satisfying
[tex]a=\dfrac{p_1}{q_1} \, b=\dfrac{p_2}{q_2}[/tex]
which implies that
[tex]a+b
=\dfrac{p_1}{q_1}+\dfrac{p_2}{q_2}
=\dfrac{p_{1}q_{2}+p_{2}q_{1}}{q_{1}q_{2}}[/tex]
By closure of [tex]\mathbb{Z}[/tex] under addition and multiplication, it follows from the assumption that [tex]p_1,p_2,q_1,q_2 \in \mathbb{Z}[/tex] that
[tex]p_{1}q_{2}+p_{2}q_{1}[/tex] and [tex]q_{1}q_{2}[/tex] are also integers.
Hence, [tex]a+b[/tex] can be expressed as the ratio of 2 integers. By definition, [tex]a+b[/tex] is rational. [tex]\square[/tex]