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Is my proof correct?
Show that, if there exists a number 0 for which x+0=x for all x∈R, and a number 0' for which x+0'=x for all x∈R, then 0=0'.
Proof by contradiction:
Assume, 0≠0'. Then,
x+0=x and x+0'=x'
Such that:
x=x-0 and x=x'-0'
Such that: x-0=x'-0'
Which is a contradiction.
Thus, 0=0'.
Homework Statement
Show that, if there exists a number 0 for which x+0=x for all x∈R, and a number 0' for which x+0'=x for all x∈R, then 0=0'.
The Attempt at a Solution
Proof by contradiction:
Assume, 0≠0'. Then,
x+0=x and x+0'=x'
Such that:
x=x-0 and x=x'-0'
Such that: x-0=x'-0'
Which is a contradiction.
Thus, 0=0'.