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The $\approx$ is used mostly in terms of numerical approximations, meaning that the values in questions are "close" to each other in whatever context one is working, and often it is less precise exactly how "close." Topologists also have a tendency to use $\approx$ for homeomorphic.
Another possible notation for the same relation is {\displaystyle A\ni x,} A\ni x, meaning "A contains x", though it is used less often. The negation of set membership is denoted by the symbol "∉". Writing {\displaystyle x\notin A} x\notin A means that "x is not an element of A".
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It's used for various things in various contexts. The one about "defined to be equal" is often rendered as ":=".
I have encountered this when referencing subsets and vector subspaces. For example, T ⊊ span(S) should mean that T is smaller than span(S)--at least from what I've gathered.
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What is the meaning of $\succ $ symbol? I have an snippet which includes this operator: (Article is about choice theory) "a complete and transitive preference $\succ$ over X" (Maybe very preliminary question but interestingly there is no explanation in the internet about this symbol) Note that $\succ$ is Unicode character U+227B.
The symbol $\sim$ does not have a set meaning across all subjects, but it is almost always used to denote an equivalence relation: a relation that is reflexive, symmetric, and transitive. Daniel Littlewood and anorton have already discussed what $\sim$ means in this instance, and we can verify that it is an equivalent relation between functions ...
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