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The symbol ≅ is used for isomorphism of objects of a category, and in particular for isomorphism of categories (which are objects of CAT). The symbol ≃ is used for equivalence of categories. At least, this is the convention used in this book and by most category theorists, although it is far from universal in mathematics at large.
A way of expressing that they are equivalent, or equal to each other by definition, is as follows: X ≡ Y. However, the symbol ≡ also denotes congruence, e.g. p2 ≡ 1 (mod6), so using the symbol for two different circumstances can cause some confusion. Thus, to denote (1), some write that X: = Y or X ≜ Y or Xdef = Y.
This is why we say $5^6$ and $1$ are "equivalent" under the modulo operation 7. and thats why we use equivalent symbol instead of equality symbol. If you are trying to write $5^6=1 (\mod 7)$ you are actually saying that n=0.
While "$≡$" denotes an equivalent statement in a mathematical equation, but definition wise it means "identical to". e.g. $3 \times 4 ≡ 4 \times 3$ (because of the commutative properties). While like others said these can be interchangeable because they do mean the same thing you usually see " $↔$ " in formal logic and " $≡$ " in ...
Lets add some more.... ∼ ∼ is "similar to" but it can also appears in equivalence relations. I suppose similarity is an equivalence relation. ≡ ≡ is "equivalent to" and appears in regularly in modular arithmetic. ≈ ≈ is aproximately. ≅ ≅ is congruence or isomorphism. And of course = = is equality. – Doug M.
You can use $\iff$, i.e. if and only if, which means that the logical statements are equivalent. If you wanted, you could define your own symbolic notation like $\equiv$ if you felt the need to for whatever reason.
1. As someone has mentioned, you use this symbol ≡ ≡, which is simply "\equiv". If you assign them any type of homework dealing with logical equivalences and solving them, then they can use Microsoft Word. Just go to Insert > Equation. There should be an Equation tab open up on the top with the other tabs.
It is also used in physics, to indicate that forces are drawed on some scale, so for example 1N Δ = 0, 1m. It is clear that a force isn't equal to some length, but they are corresponding with the same length. It is the same principle as Saphrosit says, i.e. they are by definition, and in this specific case, equal to each other.
So f(x) f (x) both denotes the function f f and the value it takes on x x. The expression f(x) = 0 f (x) = 0 can mean that for some x x, the function takes 0 0 value, and that x x needs to be found, while f(x) ≡ 0 f (x) ≡ 0 means that f f is the zero function. – Bence Racskó. Mar 5, 2015 at 11:51. Add a comment. 1 Answer.
7. What is the notation for showing that equations are equivalent after rearranging terms? For example, I sometimes solve the arc length formula. s = rθ s = r θ. for r r and write it as. r = s θ. r = s θ. When I show my work, I usually write this relationship like this: s = rθ ⇒ r = s θ. s = r θ ⇒ r = s θ. Is this the correct way to ...