◀ ▲ ▶Branches / Number-systems-arithmetics / Proof

Proof

(related to Proposition: Unique Solvability of \(a+x=b\))

• We first show that $x=b-a$ really is the solution of the equation $a+x=b.$
  • Replacing \(x\) by \((b-a)\) in the equation leads to $a+(b-a)=b.$
  • By applying the definition of subtracting real numbers we get $a+(b+(-a))=b.$
  • The commutativity of adding real numbers leads to $a+((-a)+b)=b.$
  • The associativity of adding real numbers leads to $(a+(-a))+b=b.$
  • The uniqueness of negative numbers leads to $0+b=b.$
  • The existence of zero leads to $b=b.$
  • Thus, $x=b-a$ is the solution of the equation $a+x=b$.
• Now, we show, this solution is the only solution of the equation.
  • Take any real number \(y\) with $a+y=b.$
  • Adding \((-a)\) on both sides of the equation leads to $(-a)+(a+y)=(-a)+b.$
  • By associativity of adding real numbers we get
    $((-a)+a)+y=(-a)+b.$
  • By the uniqueness of negative numbers it follows that $0+y=(-a)+b.$
  • By the existence of zero we get $y=(-a)+b.$
  • The commutativity of adding real numbers leads to $y=b+(-a).$
  • Applying the definition of subtracting real numbers we get $y=b-a.$
  • Thus, the solution $(b-a)$ is the only solution of the equation $a+x=b.$
    ∎

Mentioned in:

Parts: 1

Thank you to the contributors under CC BY-SA 4.0!

Github:

References

Bibliography

1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983