# Using implication and equality

I've searched around a quite bit for an answer to this question but I haven't found the answer. If someone can find a link to a question that has an answer that also answers this question that would be great. If not, here is my question.

I am an engineering student and for years I have thought that I was using the implication arrow ($\Rightarrow$) and the equality sign ($=$) in a correct way but recently a friend told me that I was using them incorrectly. As I understand , the equality sign means that the expressions on both sides of the sign are equal and that you can use many of them in a sequence, for example: $$x^3+2x^2+x=x(x^2+2x+1)=x(x+1)^2$$ As for the implication arrow, as far as I understand this is considered correct usage: $$x^3+2x^2+2x=x\ \Rightarrow \ x^3+2x^2+x=0 \ \Rightarrow \ x(x^2+2x+1)=0 \\ \Rightarrow x(x+1)^2=0\ \Rightarrow \ x=0 \ \lor \ x=-1$$ I assume that what I have done above is correct so here comes the part where we disagree. My friend says that I can not use a sequence of equality signs after an implication arrow, that is only one equality can follow an implication. He would say that the following usage is incorrect: $$x^3+2x^2+2x=x\ \Rightarrow \ x^3+2x^2+x=x(x^2+2x+1)=x(x+1)^2=0 \\ \Rightarrow \ x=0 \lor \ x=-1$$ or (this is closer to the case of our original dispute) $$x=5 \quad \text{ and } \quad y=10 \\ \Rightarrow x^2+y^2=25+100=125$$ I don't remember if my friend gave me a clear reason why this was wrong, but he said that his high school/junior college math teacher told him that it was. Can anyone settle this dispute for us? Is this perhaps just a matter of preference?

• $x^3+2x^2+x=x(x^2+2x+1)=x(x+1)^2$ are two identities true for all $x$ so do not directly lead to the following implication and do not depend on the previous equation. Only the equation being $=0$ does. So I think your friend's teacher's point is that the use of equalities and implications is better when less ambiguous Mar 12, 2018 at 1:13
• Your use of notation looks fine to me (though it would probably be more semantically correct to use \implies rather than \rightarrow). I understand the potential ambiguity that @Henry points out, but it doesn't seem like an issue to me. Mar 12, 2018 at 1:27

That looks right. If you want to be explicit:

$$x(x+1)^2 = 0 \implies (x = 0 \vee x = -1)$$ $$(x = 5 \wedge y = 10) \implies x^2 + y^2 = 125$$

This is logically accurate, and more than sufficient for casual explanations in precalculus.