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Is it proper to mix logical notation with set theory notation? I would like to better understand when writing out "if" or "and" is necessary.

In these examples, the use of → to denote both a conditional and a mapping seems confusing. Is using a symbol to mean two different things a conflict?

"A equals B if every element x of A is an element of B and every element x of B is an element of A"

∀x[x ∈ A ⟹ x ∈ B] ∧ ∀x[x ∈ B ⟹ x ∈ A] ⟹ A = B

"A equals B if A and B have the same elements"?

∀x[x ∈ B ⟺ x ∈ A] ⟹ A = B

Greatly appreciated,

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  • $\begingroup$ I think either of the sentences in words is better than either of the sentences in symbols.They'd be even better if you said "if and only if" or "just when". $\endgroup$
    – Ethan Bolker
    Mar 30, 2019 at 1:38
  • $\begingroup$ That makes sense - it would be clearer! Assuming the sentences were of sufficient complexity or required using symbols alone, is there a better way to phrase this? Thank you. $\endgroup$
    – WarmTaunTaun
    Mar 30, 2019 at 1:44
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    $\begingroup$ "Sufficient complexity" works the other way. The more complex the thought the more you need words, I don't know when you might be required to use symbols alone - perhaps in a computer program but not for human readers. See math.stackexchange.com/questions/2732988/… $\endgroup$
    – Ethan Bolker
    Mar 30, 2019 at 1:49
  • $\begingroup$ Please try to write mathematical formulas in mathjax. math.meta.stackexchange.com/questions/5020/… $\endgroup$
    – Bertrand Wittgenstein's Ghost
    Mar 30, 2019 at 1:53
  • $\begingroup$ Thank you for the suggestion @BertrandWittgenstein'sGhost, edited formatting to mathjax. I should add that I'm looking for clarification on what to do if two symbols conflict in meaning, such as in the example. If using logical symbols in in this manner is ever proper. $\endgroup$
    – WarmTaunTaun
    Mar 30, 2019 at 2:09

3 Answers 3

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Indeed notations coincide, but often it is easy to recognize (from the context) which meaning you intended for the arrow ($\to$).

A mapping is an object from set theory, it is a mapping from a set to a set.

The arrow you use in the sentences are not arrows from sets to sets. They're arrows from statements about sets to statements about sets. So, they definitely are not functions that map sets to sets.

If you still want to avoid this, you can use use a regular arrow ($\to$) for mappings and double arrow ($\Rightarrow$) for implication between statements.

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    $\begingroup$ There's also $\mapsto$ , although that is sometimes reserved for maps element to element rather than set to set. $\endgroup$
    – Graham Kemp
    Mar 30, 2019 at 7:38
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Many symbols in math are reused and need context to distinguish the possibilities. Rightward arrows can be mappings or can be implies or probably some other things I haven't thought of right now. In this case there are no mappings in sight, so I have no doubt the arrows are implies. If you ask for implies from MathJax you get $\implies$. If I had a problem that involved both mappings and implies, I would define different arrows in a preface.

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  • $\begingroup$ The example is based on lecture notes from the University of Texas link: "A set A is a subset of set B, denoted A ⊆ B, if every element x of A is also an element of B. That is, A ⊆ B if ∀x(x ∈ A → x ∈ B)." They were using → as an imply, not a mapping? I must have misunderstood. In this case, should I edit the post's example to better reflect the question? $\endgroup$
    – WarmTaunTaun
    Mar 30, 2019 at 2:50
  • $\begingroup$ Yes, that use is clearly implies. To have a mapping, you need a set on each side of the arrow, but here there is a sentence. Once you have sentences, you connect them with logical connectives. $\endgroup$
    – Ross Millikan
    Mar 30, 2019 at 3:04
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This the definition of set equality:

$$\forall A,B[A=B\iff (A\subseteq B\land B\subseteq A)]$$

Which translates to: for any two sets, they are equal if and only if they are subsets of one another (or, as you said they have exactly the same elements).

I hope that answers your question.

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