Discrete Math Question Prove the following statement: "The sum of any two rational

sagnuhh 2021-08-20 Answered
Discrete Math Question
Prove the following statement:
"The sum of any two rational numbers is rational."

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Expert Answer

Arham Warner
Answered 2021-08-21 Author has 13600 answers

Step 1
Concept Used:
A rational number is the number that can be written in the form \(\displaystyle{\frac{{{a}}}{{{b}}}}\) where , a and b are integers and \(\displaystyle{b}\ne{}{0}\)
Step 2
Proof:
We want to prove that the sum of two rational numbers is rational
Let \(\displaystyle{x}={\frac{{{a}}}{{{b}}}}\) where \(\displaystyle{b}\ne{0},{a},{b}\) are integers
and \(\displaystyle{y}={\frac{{{a}}}{{{b}}}}\) where \(\displaystyle{d}\ne{0},{c},{d}\) are integers
consider
\(\displaystyle{x}+{y}={\frac{{{a}}}{{{b}}}}+{\frac{{{c}}}{{{d}}}}\)
Since a,b,c,d are integers then \(\displaystyle{a}{d}+{b}{c}\) and bd is integer and \(\displaystyle{b}\ne{}{0},{d}\ne{}{0}\Rightarrow{b}{d}\ne{}{0}\)
\(\displaystyle\Rightarrow{x}+{y}\) rational
hence proved.

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