This is from a text book:-
"The general form of a linear equation in two variables is $ax + by + c = 0$ or, $ax + by = c$ where $a, b, c$ are real numbers such that $a ≠ 0$, $b≠0$ and $x, y$ are variables.
(we often denote the condition $a$ and $b$ are not both zero by $a^2+b^2≠0$.)"
I don’t understand this last condition.
How can we say that $a^2+b^2≠0$ represents the condition that $a$ and $b$ are not both zero.
Let $a = 0, b = 1$, then also this condition fulfills.
Any help?
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Shekhar Malhotra
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Generally words "together", "both" and comma sign "," is used for logical operation AND (conjunction), denoted by $x \land y$. So $$(a \ne 0, b \ne 0) \Leftrightarrow (a \ne 0 \land b \ne 0) \Leftrightarrow (a^2+b^2 \ne 0)$$
zkutch
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$a$ and $b$ are both zero $\iff$ $a=b=0$, so
$a$ and $b$ are NOT both zero $\iff$ at least one of $a,b$ is not $0$
which is equivalent to $a^2+b^2=0$ in the case that $a,b$ are both real numbers.
Answering to your comment, yes, you are right. Maybe a better way to write in an inequality is $(a,b)\neq(0,0)$ instead of "$a\neq0,b\neq0$"
Brian Cheung
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(a,b)≠(0,0); I am not familiar with this notation. What does it mean? – Shekhar Malhotra Jun 16 '20 at 08:14
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@ShekharMalhotra Geometrically you can think of it as 'the point (a,b) is not the same point as the origin'. Algebraically, $a\neq0$ AND $b\neq0$. In normal English words, '$a,b$ are not both zero. – Brian Cheung Jun 16 '20 at 08:31
This statement reflects the fact that both $a$ and $b$ are non zero.
[B] $a^2+b^2\neq 0$*.
This statement reflects the fact that one of $a$ and $b$ can be zero.
Isn't there a contradiction? – Shekhar Malhotra Jun 16 '20 at 05:58