Working on proof writing, and I need to prove
$$(-x)y=-(xy)$$
using the axioms of the real numbers. I know that this is equivalent to saying that the additive inverse of $xy$ is $(-x)y$ but I am unsure how to prove it.
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$\begingroup$ One thing you can do to start a proof like this is look at what other structures have this property. For example, this theorem holds for matrices, so you can infer that any axiom of real number which doesn't hold for matrices (such as multiplicative commutivity and the order properties) won't be helpful in proving the theorem. It holds in modular arithmetic in a non-prime base, so the unique inverse of multiplication isn't helpful either. $\endgroup$– DanielVAug 21, 2016 at 22:10
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4 Answers
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You could prove that $(-x)y$ and $-(xy)$ are both the additive inverse of $xy$. Then use its uniqueness ($5^{th}$ axiom).
$$xy + (-x)y = (x+(-x))y = 0y = 0$$
Notice that $0y = 0$, because : $$0y = (0+0)y = 0y + 0y$$ and if you add the additive inverse of $0y$ to both of the sides you come up with : $$(-0y) + 0y = (-0y) + 0y + 0y$$ i.e. $$0 = 0y$$
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$\begingroup$ how can we prove that $xy+(-x)y=0$? if we don't have $0.y=0$ we need this axiom otherwise we have to prove it $\endgroup$– ElaqqadApr 3, 2015 at 12:31
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$\begingroup$ @Elaqqad $0y = 0$ can be easily proven. I'll add it to my answer. $\endgroup$– brickApr 3, 2015 at 12:33
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1$\begingroup$ Ok now it's fine , because I think that the main part is $0y=y$ $\endgroup$– ElaqqadApr 3, 2015 at 12:40
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Associative property: $$(xy)z=x(yz)$$ Then: $$(-x)y=-(xy)\Longrightarrow(-1x)y=-1(xy)$$
When: $z=-1$
answered Aug 21, 2016 at 21:23
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1$\begingroup$ That still leaves it to prove that $(-1)x = -x$, assuming $-$ is defined as the inverse of addition. $\endgroup$– DanielVAug 21, 2016 at 21:39
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$\begingroup$ To establish that $(-1)x = -x$, add $1x$ to both sides of the equation (although a real proof would probably use the existence of an inverse rather than the invertibility of addition, since invertibility isn't an axiom). $\endgroup$– DanielVAug 21, 2016 at 22:00
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First start by proving : -x = (-1)x. Proof: x+(-1)x = (1+(-1))x = 0x = 0. The proposition follows from uniqnes off negative off a number. Now you can use this to prove your statement. Proof: (−x)y = ((-1)x)y = (-1)(xy) = -xy
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$$ 0 = 0 $$ $$ 0y = 0 $$ $$ (x+(-x))y = 0 $$ Distributive: $$ xy + (-x)y = 0 $$ Add $ −(xy) $ to both sides: $$ - (xy) + (xy + (-x)y) = -(xy) + 0 $$ Associative of +: $$ (- (xy) + xy) + (-x)y = -(xy) $$ $$ (-x)y = -(xy) $$
The master key is the distribution.
The axiom of distribution implies precisely what we want to demonstrate.
Without distribution, nothing links the operation + with the operation *. You would only have two groups on the same set.
