symbol for a 'is the same as saying' b or a 'can be rewritten as' b
Hi everyone I searched a bit and couldn't find anything quite specifically for what i'm looking for. Am curious to know what is the best way to symbolically write or convey the idea of 'I'm rewriting this as' or 'lets look at it this way'.
The logically equivalent symbol $ equiv $ comes closest I think but is it appropriate to use it in cases like: $$ 2a leq 2^n equiv a+a leq 2 cdot 2^{n-1}$$
In this case is it just preferable to use brackets if I want to expand out the inequality like this ie:
$$ (2a leq 2^n ) = (a + a leq 2 cdot 2^{n-1}) $$
just doesn't really seem right. And while = typically works I have found in some cases it can cause a bit of ambiguity I prefer to avoid especially when starting out on working out problems. Another example would be something like:
$$ (g circ f) (x) equiv g(f(x)) $$
Again here equals doesn't seem quite right but but neither does logically equivalent as there's no truth value involved - I'm just trying to frame the meaning different and remind myself or reader how we're going to approach it.
Hopefully question makes sense its just I haven't found quite explaining this and am just wondering if there is a standard or common way of doing this. I just like the idea of having something concise that means 'can be rewritten as' etc. Thanks.
discrete-mathematics logic
asked Nov 26 at 0:24
backslash
165
add a comment |
Hi everyone I searched a bit and couldn't find anything quite specifically for what i'm looking for. Am curious to know what is the best way to symbolically write or convey the idea of 'I'm rewriting this as' or 'lets look at it this way'.
The logically equivalent symbol $ equiv $ comes closest I think but is it appropriate to use it in cases like: $$ 2a leq 2^n equiv a+a leq 2 cdot 2^{n-1}$$
In this case is it just preferable to use brackets if I want to expand out the inequality like this ie:
$$ (2a leq 2^n ) = (a + a leq 2 cdot 2^{n-1}) $$
just doesn't really seem right. And while = typically works I have found in some cases it can cause a bit of ambiguity I prefer to avoid especially when starting out on working out problems. Another example would be something like:
$$ (g circ f) (x) equiv g(f(x)) $$
Again here equals doesn't seem quite right but but neither does logically equivalent as there's no truth value involved - I'm just trying to frame the meaning different and remind myself or reader how we're going to approach it.
Hopefully question makes sense its just I haven't found quite explaining this and am just wondering if there is a standard or common way of doing this. I just like the idea of having something concise that means 'can be rewritten as' etc. Thanks.
discrete-mathematics logic
asked Nov 26 at 0:24
backslash
165
2
I feel like the "if and only if" symbol is probably the most valid? For example: $$a + a = 2 ;;; Leftrightarrow ;;; 2a = 2$$ but I'm not sure about it. I feel like I've seen it used in that way before, and from a logical standpoint in the case of "just rewriting" or whatever to make certain steps more clear I would say it's valid. Granted I'm no expert on the matter and my experiences are anecdotal.
– Eevee Trainer
Nov 26 at 0:27
2
Though on the matter of composition of functions I think "$=$" is the most valid. You could also use "$:=$" since that basically means "is defined by."
– Eevee Trainer
Nov 26 at 0:29
Yeah these are the two I came across while searching closest to what I'm after. The 'is defined by' notation works in some cases def (composition of functions like you mentioned its pretty good). Others though( $$ 2+2 := 4 $$ and $$ 2^{n} leftrightarrow 2 cdot 2^{n-1} $$ its a little awkward - Though to be fair, in these cases = should be sufficient). They are close but was still looking for something a bit more free of context / truthiness - kind of a universal 'lets look at it this way'. May just have to use a combination of them or a couple words explaining it. Thanks for input!
– backslash
Nov 26 at 21:22
You're basically reaffirming my own thoughts about it - where those two symbols are close but not exact the same and depends on the scenario. I'm going to try to work with these and look into more of the examples where I felt these weren't ideal and see what I come up with - it may be possible in each of those cases by changing the order or arrangement of the proofs / explanations that I can utilize accurately := ('is defined by') and $ leftrightarrow $ to get what I'm after.
– backslash
Nov 26 at 21:32
add a comment |
Hi everyone I searched a bit and couldn't find anything quite specifically for what i'm looking for. Am curious to know what is the best way to symbolically write or convey the idea of 'I'm rewriting this as' or 'lets look at it this way'.
The logically equivalent symbol $ equiv $ comes closest I think but is it appropriate to use it in cases like: $$ 2a leq 2^n equiv a+a leq 2 cdot 2^{n-1}$$
In this case is it just preferable to use brackets if I want to expand out the inequality like this ie:
$$ (2a leq 2^n ) = (a + a leq 2 cdot 2^{n-1}) $$
just doesn't really seem right. And while = typically works I have found in some cases it can cause a bit of ambiguity I prefer to avoid especially when starting out on working out problems. Another example would be something like:
$$ (g circ f) (x) equiv g(f(x)) $$
Again here equals doesn't seem quite right but but neither does logically equivalent as there's no truth value involved - I'm just trying to frame the meaning different and remind myself or reader how we're going to approach it.
Hopefully question makes sense its just I haven't found quite explaining this and am just wondering if there is a standard or common way of doing this. I just like the idea of having something concise that means 'can be rewritten as' etc. Thanks.
discrete-mathematics logic
asked Nov 26 at 0:24
backslash
165
Hi everyone I searched a bit and couldn't find anything quite specifically for what i'm looking for. Am curious to know what is the best way to symbolically write or convey the idea of 'I'm rewriting this as' or 'lets look at it this way'.
The logically equivalent symbol $ equiv $ comes closest I think but is it appropriate to use it in cases like: $$ 2a leq 2^n equiv a+a leq 2 cdot 2^{n-1}$$
In this case is it just preferable to use brackets if I want to expand out the inequality like this ie:
$$ (2a leq 2^n ) = (a + a leq 2 cdot 2^{n-1}) $$
just doesn't really seem right. And while = typically works I have found in some cases it can cause a bit of ambiguity I prefer to avoid especially when starting out on working out problems. Another example would be something like:
$$ (g circ f) (x) equiv g(f(x)) $$
Again here equals doesn't seem quite right but but neither does logically equivalent as there's no truth value involved - I'm just trying to frame the meaning different and remind myself or reader how we're going to approach it.
Hopefully question makes sense its just I haven't found quite explaining this and am just wondering if there is a standard or common way of doing this. I just like the idea of having something concise that means 'can be rewritten as' etc. Thanks.
discrete-mathematics logic
discrete-mathematics logic
asked Nov 26 at 0:24
backslash
165
asked Nov 26 at 0:24
backslash
165
asked Nov 26 at 0:24
backslash
165
asked Nov 26 at 0:24
backslash
165
asked Nov 26 at 0:24
backslash
165
165
2
I feel like the "if and only if" symbol is probably the most valid? For example: $$a + a = 2 ;;; Leftrightarrow ;;; 2a = 2$$ but I'm not sure about it. I feel like I've seen it used in that way before, and from a logical standpoint in the case of "just rewriting" or whatever to make certain steps more clear I would say it's valid. Granted I'm no expert on the matter and my experiences are anecdotal.
– Eevee Trainer
Nov 26 at 0:27
2
Though on the matter of composition of functions I think "$=$" is the most valid. You could also use "$:=$" since that basically means "is defined by."
– Eevee Trainer
Nov 26 at 0:29
Yeah these are the two I came across while searching closest to what I'm after. The 'is defined by' notation works in some cases def (composition of functions like you mentioned its pretty good). Others though( $$ 2+2 := 4 $$ and $$ 2^{n} leftrightarrow 2 cdot 2^{n-1} $$ its a little awkward - Though to be fair, in these cases = should be sufficient). They are close but was still looking for something a bit more free of context / truthiness - kind of a universal 'lets look at it this way'. May just have to use a combination of them or a couple words explaining it. Thanks for input!
– backslash
Nov 26 at 21:22
You're basically reaffirming my own thoughts about it - where those two symbols are close but not exact the same and depends on the scenario. I'm going to try to work with these and look into more of the examples where I felt these weren't ideal and see what I come up with - it may be possible in each of those cases by changing the order or arrangement of the proofs / explanations that I can utilize accurately := ('is defined by') and $ leftrightarrow $ to get what I'm after.
– backslash
Nov 26 at 21:32
add a comment |
2
I feel like the "if and only if" symbol is probably the most valid? For example: $$a + a = 2 ;;; Leftrightarrow ;;; 2a = 2$$ but I'm not sure about it. I feel like I've seen it used in that way before, and from a logical standpoint in the case of "just rewriting" or whatever to make certain steps more clear I would say it's valid. Granted I'm no expert on the matter and my experiences are anecdotal.
– Eevee Trainer
Nov 26 at 0:27
2
Though on the matter of composition of functions I think "$=$" is the most valid. You could also use "$:=$" since that basically means "is defined by."
– Eevee Trainer
Nov 26 at 0:29
Yeah these are the two I came across while searching closest to what I'm after. The 'is defined by' notation works in some cases def (composition of functions like you mentioned its pretty good). Others though( $$ 2+2 := 4 $$ and $$ 2^{n} leftrightarrow 2 cdot 2^{n-1} $$ its a little awkward - Though to be fair, in these cases = should be sufficient). They are close but was still looking for something a bit more free of context / truthiness - kind of a universal 'lets look at it this way'. May just have to use a combination of them or a couple words explaining it. Thanks for input!
– backslash
Nov 26 at 21:22
You're basically reaffirming my own thoughts about it - where those two symbols are close but not exact the same and depends on the scenario. I'm going to try to work with these and look into more of the examples where I felt these weren't ideal and see what I come up with - it may be possible in each of those cases by changing the order or arrangement of the proofs / explanations that I can utilize accurately := ('is defined by') and $ leftrightarrow $ to get what I'm after.
– backslash
Nov 26 at 21:32
2
2
I feel like the "if and only if" symbol is probably the most valid? For example: $$a + a = 2 ;;; Leftrightarrow ;;; 2a = 2$$ but I'm not sure about it. I feel like I've seen it used in that way before, and from a logical standpoint in the case of "just rewriting" or whatever to make certain steps more clear I would say it's valid. Granted I'm no expert on the matter and my experiences are anecdotal.
– Eevee Trainer
Nov 26 at 0:27
I feel like the "if and only if" symbol is probably the most valid? For example: $$a + a = 2 ;;; Leftrightarrow ;;; 2a = 2$$ but I'm not sure about it. I feel like I've seen it used in that way before, and from a logical standpoint in the case of "just rewriting" or whatever to make certain steps more clear I would say it's valid. Granted I'm no expert on the matter and my experiences are anecdotal.
– Eevee Trainer
Nov 26 at 0:27
2
2
Though on the matter of composition of functions I think "$=$" is the most valid. You could also use "$:=$" since that basically means "is defined by."
– Eevee Trainer
Nov 26 at 0:29
Though on the matter of composition of functions I think "$=$" is the most valid. You could also use "$:=$" since that basically means "is defined by."
– Eevee Trainer
Nov 26 at 0:29
Yeah these are the two I came across while searching closest to what I'm after. The 'is defined by' notation works in some cases def (composition of functions like you mentioned its pretty good). Others though( $$ 2+2 := 4 $$ and $$ 2^{n} leftrightarrow 2 cdot 2^{n-1} $$ its a little awkward - Though to be fair, in these cases = should be sufficient). They are close but was still looking for something a bit more free of context / truthiness - kind of a universal 'lets look at it this way'. May just have to use a combination of them or a couple words explaining it. Thanks for input!
– backslash
Nov 26 at 21:22
Yeah these are the two I came across while searching closest to what I'm after. The 'is defined by' notation works in some cases def (composition of functions like you mentioned its pretty good). Others though( $$ 2+2 := 4 $$ and $$ 2^{n} leftrightarrow 2 cdot 2^{n-1} $$ its a little awkward - Though to be fair, in these cases = should be sufficient). They are close but was still looking for something a bit more free of context / truthiness - kind of a universal 'lets look at it this way'. May just have to use a combination of them or a couple words explaining it. Thanks for input!
– backslash
Nov 26 at 21:22
You're basically reaffirming my own thoughts about it - where those two symbols are close but not exact the same and depends on the scenario. I'm going to try to work with these and look into more of the examples where I felt these weren't ideal and see what I come up with - it may be possible in each of those cases by changing the order or arrangement of the proofs / explanations that I can utilize accurately := ('is defined by') and $ leftrightarrow $ to get what I'm after.
– backslash
Nov 26 at 21:32
You're basically reaffirming my own thoughts about it - where those two symbols are close but not exact the same and depends on the scenario. I'm going to try to work with these and look into more of the examples where I felt these weren't ideal and see what I come up with - it may be possible in each of those cases by changing the order or arrangement of the proofs / explanations that I can utilize accurately := ('is defined by') and $ leftrightarrow $ to get what I'm after.
– backslash
Nov 26 at 21:32
add a comment |
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2
I feel like the "if and only if" symbol is probably the most valid? For example: $$a + a = 2 ;;; Leftrightarrow ;;; 2a = 2$$ but I'm not sure about it. I feel like I've seen it used in that way before, and from a logical standpoint in the case of "just rewriting" or whatever to make certain steps more clear I would say it's valid. Granted I'm no expert on the matter and my experiences are anecdotal.
– Eevee Trainer
Nov 26 at 0:27
2
Though on the matter of composition of functions I think "$=$" is the most valid. You could also use "$:=$" since that basically means "is defined by."
– Eevee Trainer
Nov 26 at 0:29
Yeah these are the two I came across while searching closest to what I'm after. The 'is defined by' notation works in some cases def (composition of functions like you mentioned its pretty good). Others though( $$ 2+2 := 4 $$ and $$ 2^{n} leftrightarrow 2 cdot 2^{n-1} $$ its a little awkward - Though to be fair, in these cases = should be sufficient). They are close but was still looking for something a bit more free of context / truthiness - kind of a universal 'lets look at it this way'. May just have to use a combination of them or a couple words explaining it. Thanks for input!
– backslash
Nov 26 at 21:22
You're basically reaffirming my own thoughts about it - where those two symbols are close but not exact the same and depends on the scenario. I'm going to try to work with these and look into more of the examples where I felt these weren't ideal and see what I come up with - it may be possible in each of those cases by changing the order or arrangement of the proofs / explanations that I can utilize accurately := ('is defined by') and $ leftrightarrow $ to get what I'm after.
– backslash
Nov 26 at 21:32