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Let $a, b, c \in \mathbb{R}$. Prove if $a + b = a$ then $b = 0$.

Suppose that $a + b = a$. Then $a + b - a = a - a = 0 = b$ by the inverses law for addition. By the Identity law for addition it follows that $a + 0 = a$, and it follows by the Identity law for addition that if $b = 0$, then $0 + b = b = 0$, therefore $b = 0$.

Is it redundant to use the identity law for addition for a and b? Does this seem correct?

Michael Hardy
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middleRingRider
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  • It's not redundant. – Vladimir Vargas Nov 09 '14 at 01:20
  • Here's an oddity about MathJax: As you'd expect, a hyphen in math tags looks like a minus sign and one not in math tags looks like a hyphen, thus: $a-a$ versus $a$-$a$. BUT an actual minus sign in math tags looks like neither of the above, but rather like $a{-}a$. (I edited accordingly above.) ${}\qquad{}$ – Michael Hardy Nov 09 '14 at 01:33
  • Your second sentence (Stripped of some specifics: “By a law, it follows that ..., and it follows by another law, that if $b=0$ then $q$ therefore $b=0$”) is a bit wordy, and arguably not valid. Once you assume the if-part of the theorem, you must conclude the then-part (which is $b=0$). The red flag is that part of your demonstration contains “if $b=0$.” A statement of the form “if $b=0$” doesn’t belong in an argument that purports to conclude that $b=0$. – Steve Kass Nov 09 '14 at 01:44

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