My teacher claims that when an equation in variables $x_1,x_2,\ldots,x_n$ has no solutions, you should denote this fact with $(x_1,x_2,\ldots,x_n)\in\varnothing$.
An empty set can't have an element in it, so this can't be right.
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3$\begingroup$ Your teacher's idiosyncratic notation means exactly what is desired, namely that there are no elements in the set of all solutions. $\endgroup$– vadim123Nov 20, 2014 at 19:43
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$\begingroup$ To be an element of an empty set is rather strange. We just had to write down $\varnothing$ for no solution but we were not allowed to write the {} symbols around it. $\endgroup$– imranfatNov 20, 2014 at 19:44
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5$\begingroup$ @imranfat The sets $\varnothing$ and $\{\varnothing\}$ are different. One of them has an element, and the other doesn't. $\endgroup$– ArthurNov 20, 2014 at 19:45
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$\begingroup$ "should denote"? I'd say "can denote". It's not wrong, but it's not notation that is used for this very often. $\endgroup$– Simon SNov 20, 2014 at 19:46
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2$\begingroup$ An equation (or a system of equations) in so many variables has a solution set $S$ which is a subset of the universe $\Omega$ for which the equations make sense. When there are no solutions we write $S=\emptyset$. A statement as $(x_1,x_2,\ldots, x_n)\in\emptyset$ seems pretty forlorn to me. $\endgroup$– Christian BlatterNov 20, 2014 at 20:22
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4 Answers
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You can form the set $L$ of all solutions. So iff $(x_1, \cdots, x_n)$ is a solution, then $(x_1, \cdots, x_n)\in L$. Now, if there are no solutions, then $L = \varnothing$. Insert that, and you get $(x_1, \cdots, x_n)\in \varnothing$.
In other words, the sequence $(x_1, \cdots, x_n)$ is a solution to an impossible set of equations iff $(x_1, \cdots, x_n)\in \varnothing$.
However, saying that "$(x_1, \cdots, x_n)\in \varnothing$ means there are no solutions" is taking it a bit far. I would rather say "$L = \varnothing$ means there are no solutions".
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You are correct that the empty set is the set with no members. The solution set of an equation is, however defined as the set of all solutions that satisfy the equation and since that set has no members the solution set is empty.
Thus, saying that the equation has no solution, and saying that the solution set of the equation is empty mean the same thing.
Recall that the solution set is the set of all values that turn an open sentence into a true statement.
Thus, the membership symbol is an abbreviation for the fact that the variable can be replaced by a value that is "in", "belongs to", or "is a member of" the set. If the set is empty, then the variable can not be replaced by any value. This is what it means for an equation to have "no solution", there is no value that when substituted for the variable will result in a true statement.
I agree with you that the notation does lead to some confusion when dealing with the special case of the solution set being empty. In this case "what the variable can be replaced by belongs to" a set with no members. But, if you like, you can simply say "No solution." Note that the meaning here is still the value that the variable can be replaced by belongs to the replacement set, which happens to be empty so there is no value.
variable: A symbol used to represent any member of a given set.
domain of a variable: The set whose members may serve as replacements for the variable; also called replacement set.
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$\begingroup$ Of course, surely the set of all solutions is empty in this case, but how can an object exist in a fully empty set? If you think this is correct notation, then should $A\in\varnothing$, where $A=\{A\}$ also be correct? In both cases, the object is impossible and if you agree with my teacher's claim, then you agree with this claim, thus overall agreeing that all impossible objects exist in the empty set. $\endgroup$ Nov 20, 2014 at 21:31
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$\begingroup$ Again, yes, the solution set is empty. I don't know how you defined an object, but let it be something that can exist in a set (which is just a collection). $(x_1,\ldots,x_n)$ is an object and thus should be in a set. But $(x_1,\ldots,x_n)\in\varnothing$ implies it is not able to be a member of a set and therefore it is not a possible object. $(x_1,\ldots,x_n)\in\varnothing$ thus implies an impossible object is in the empty set. But that means that all impossible objects exist in the empty set. You agree with my teacher's claim iff you agree with this claim, unless I'm missing something. $\endgroup$ Nov 20, 2014 at 21:52
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$\begingroup$ It simply means $\not\exists(x_1,\ldots,x_n)(f(x_1,\ldots,x_n)=0)$, where $f(x_1,\ldots,x_n)=0$ is the equation we're trying to solve. In your case, $f(x)=x+2-x-1=1$ and so it is true that $\not\exists x(f(x)=0)$, since $\forall x\in\mathbb R, f(x)=1$. $\endgroup$ Nov 20, 2014 at 22:04
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$\begingroup$ $x$ is an object. $\varnothing$, or $\{\}$, is a set with no objects. $\in$ means 'is in'. Therefore, $x\in\varnothing$ denotes '$x$ is an object in a set with no objects'. $\endgroup$ Nov 20, 2014 at 22:17
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$\begingroup$ If you will, let us call it 'something' instead. $x$ is something. $\{\}$ contains nothing. How can something be where nothing is? $\endgroup$ Nov 20, 2014 at 22:23
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I would write $card\{X \mid f(X) = 0\} = 0 $ where $card\{S\}$ is the number of elements in a set $S$ and $X$ is a vector $(x_1, x_2, ..., x_n)$.
Another way would be $\{X \mid f(X) = 0\} =\varnothing $.
answered Nov 20, 2014 at 20:12
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$\begingroup$ I didn't know that $f(\vec{a})=f(a_1,a_2,\ldots,a_n)$, where $\vec{a}=\{a_1,a_2,\ldots,a_n\}$. $\endgroup$ Nov 20, 2014 at 20:17
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$\begingroup$ That is a convenient and standard way of writing vectors. $\endgroup$ Nov 20, 2014 at 20:22
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$\begingroup$ You could also simply write $\not\exists (x_1,\ldots,x_n)\left( f(x_1,\ldots,x_n)=0 \right)$. $\endgroup$ Nov 20, 2014 at 20:47
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A more efficient way to denote that the equation has no solution, is by a Boolean "false", "F", or "$\bot$", consistent with your observation that the empty set cannot have any elements.
