Using terms “necessary” and “sufficient” when writing proofs for logical equivalence?
Often times in mathematics writing, authors prove the two sides of the implication by showing a proof of necessity and a proof of sufficiency, and the proof takes the following template:
Proposition: $A$ if and only if $B$
Proof.
(Necessity) [... proof of $A implies B$ here ...]
(Sufficiency) [... proof of $B implies A$ here ...]
My question is: on what basis does the writer use (Sufficiency) used to mean $B implies A$ and (Necessity) to mean $A implies B$?
The choice seems rather arbitrary, as one can write the implication $A implies B$ as either "$A$ is sufficient for $B$" or as "$B$ is necessary for $A$".
Is there a consistent pattern where writers typically choose "sufficient" and "necessary" both in reference to $A$ because $A$ appeared first in the statement of the Proposition?
logic proof-writing notation article-writing
asked Nov 29 at 4:40
jesterII
1,19621226
|
show 2 more comments
Often times in mathematics writing, authors prove the two sides of the implication by showing a proof of necessity and a proof of sufficiency, and the proof takes the following template:
Proposition: $A$ if and only if $B$
Proof.
(Necessity) [... proof of $A implies B$ here ...]
(Sufficiency) [... proof of $B implies A$ here ...]
My question is: on what basis does the writer use (Sufficiency) used to mean $B implies A$ and (Necessity) to mean $A implies B$?
The choice seems rather arbitrary, as one can write the implication $A implies B$ as either "$A$ is sufficient for $B$" or as "$B$ is necessary for $A$".
Is there a consistent pattern where writers typically choose "sufficient" and "necessary" both in reference to $A$ because $A$ appeared first in the statement of the Proposition?
logic proof-writing notation article-writing
asked Nov 29 at 4:40
jesterII
1,19621226
2
It is often the case (I do because it is better English) that authors DO write "For A to be the case, it is necessary and sufficient, that B should hold as well." So, I would never ever write "if and only if" or the uglier version "iff."
– Will M.
Nov 29 at 4:43
1
In my experience, authors always keep the ordering of $A$ and $B$. That is, proof of sufficiency means $A implies B$ while proof of necessity means $A impliedby B$. The arbitrariness is only there when you switch the placement of $A$ and $B$.
– Ron
Nov 29 at 4:48
1
I am confused. In the highlighted part you say that the (Necessity) proof is the proof of $A Rightarrow B$, but below that, you say that that is the (Sufficiency) proof?
– Bram28
Nov 29 at 4:49
I agree, I have encountered this specific one and has left me confused as well. I don't think that there is a consistent pattern and even bad/wrong English, as @WillM suggests, is constantly reproduced by us non-native English speakers just because we see it in some paper and think that therefore it must also be correct.
– Jimmy R.
Nov 29 at 4:50
See also the post Conditional Statements: “only if”
– Mauro ALLEGRANZA
Nov 29 at 7:14
|
show 2 more comments
Often times in mathematics writing, authors prove the two sides of the implication by showing a proof of necessity and a proof of sufficiency, and the proof takes the following template:
Proposition: $A$ if and only if $B$
Proof.
(Necessity) [... proof of $A implies B$ here ...]
(Sufficiency) [... proof of $B implies A$ here ...]
My question is: on what basis does the writer use (Sufficiency) used to mean $B implies A$ and (Necessity) to mean $A implies B$?
The choice seems rather arbitrary, as one can write the implication $A implies B$ as either "$A$ is sufficient for $B$" or as "$B$ is necessary for $A$".
Is there a consistent pattern where writers typically choose "sufficient" and "necessary" both in reference to $A$ because $A$ appeared first in the statement of the Proposition?
logic proof-writing notation article-writing
asked Nov 29 at 4:40
jesterII
1,19621226
Often times in mathematics writing, authors prove the two sides of the implication by showing a proof of necessity and a proof of sufficiency, and the proof takes the following template:
Proposition: $A$ if and only if $B$
Proof.
(Necessity) [... proof of $A implies B$ here ...]
(Sufficiency) [... proof of $B implies A$ here ...]
My question is: on what basis does the writer use (Sufficiency) used to mean $B implies A$ and (Necessity) to mean $A implies B$?
The choice seems rather arbitrary, as one can write the implication $A implies B$ as either "$A$ is sufficient for $B$" or as "$B$ is necessary for $A$".
Is there a consistent pattern where writers typically choose "sufficient" and "necessary" both in reference to $A$ because $A$ appeared first in the statement of the Proposition?
logic proof-writing notation article-writing
logic proof-writing notation article-writing
asked Nov 29 at 4:40
jesterII
1,19621226
asked Nov 29 at 4:40
jesterII
1,19621226
edited Nov 30 at 14:37
asked Nov 29 at 4:40
jesterII
1,19621226
asked Nov 29 at 4:40
jesterII
1,19621226
asked Nov 29 at 4:40
jesterII
1,19621226
1,19621226
2
It is often the case (I do because it is better English) that authors DO write "For A to be the case, it is necessary and sufficient, that B should hold as well." So, I would never ever write "if and only if" or the uglier version "iff."
– Will M.
Nov 29 at 4:43
1
In my experience, authors always keep the ordering of $A$ and $B$. That is, proof of sufficiency means $A implies B$ while proof of necessity means $A impliedby B$. The arbitrariness is only there when you switch the placement of $A$ and $B$.
– Ron
Nov 29 at 4:48
1
I am confused. In the highlighted part you say that the (Necessity) proof is the proof of $A Rightarrow B$, but below that, you say that that is the (Sufficiency) proof?
– Bram28
Nov 29 at 4:49
I agree, I have encountered this specific one and has left me confused as well. I don't think that there is a consistent pattern and even bad/wrong English, as @WillM suggests, is constantly reproduced by us non-native English speakers just because we see it in some paper and think that therefore it must also be correct.
– Jimmy R.
Nov 29 at 4:50
See also the post Conditional Statements: “only if”
– Mauro ALLEGRANZA
Nov 29 at 7:14
|
show 2 more comments
2
It is often the case (I do because it is better English) that authors DO write "For A to be the case, it is necessary and sufficient, that B should hold as well." So, I would never ever write "if and only if" or the uglier version "iff."
– Will M.
Nov 29 at 4:43
1
In my experience, authors always keep the ordering of $A$ and $B$. That is, proof of sufficiency means $A implies B$ while proof of necessity means $A impliedby B$. The arbitrariness is only there when you switch the placement of $A$ and $B$.
– Ron
Nov 29 at 4:48
1
I am confused. In the highlighted part you say that the (Necessity) proof is the proof of $A Rightarrow B$, but below that, you say that that is the (Sufficiency) proof?
– Bram28
Nov 29 at 4:49
I agree, I have encountered this specific one and has left me confused as well. I don't think that there is a consistent pattern and even bad/wrong English, as @WillM suggests, is constantly reproduced by us non-native English speakers just because we see it in some paper and think that therefore it must also be correct.
– Jimmy R.
Nov 29 at 4:50
See also the post Conditional Statements: “only if”
– Mauro ALLEGRANZA
Nov 29 at 7:14
2
2
It is often the case (I do because it is better English) that authors DO write "For A to be the case, it is necessary and sufficient, that B should hold as well." So, I would never ever write "if and only if" or the uglier version "iff."
– Will M.
Nov 29 at 4:43
It is often the case (I do because it is better English) that authors DO write "For A to be the case, it is necessary and sufficient, that B should hold as well." So, I would never ever write "if and only if" or the uglier version "iff."
– Will M.
Nov 29 at 4:43
1
1
In my experience, authors always keep the ordering of $A$ and $B$. That is, proof of sufficiency means $A implies B$ while proof of necessity means $A impliedby B$. The arbitrariness is only there when you switch the placement of $A$ and $B$.
– Ron
Nov 29 at 4:48
In my experience, authors always keep the ordering of $A$ and $B$. That is, proof of sufficiency means $A implies B$ while proof of necessity means $A impliedby B$. The arbitrariness is only there when you switch the placement of $A$ and $B$.
– Ron
Nov 29 at 4:48
1
1
I am confused. In the highlighted part you say that the (Necessity) proof is the proof of $A Rightarrow B$, but below that, you say that that is the (Sufficiency) proof?
– Bram28
Nov 29 at 4:49
I am confused. In the highlighted part you say that the (Necessity) proof is the proof of $A Rightarrow B$, but below that, you say that that is the (Sufficiency) proof?
– Bram28
Nov 29 at 4:49
I agree, I have encountered this specific one and has left me confused as well. I don't think that there is a consistent pattern and even bad/wrong English, as @WillM suggests, is constantly reproduced by us non-native English speakers just because we see it in some paper and think that therefore it must also be correct.
– Jimmy R.
Nov 29 at 4:50
I agree, I have encountered this specific one and has left me confused as well. I don't think that there is a consistent pattern and even bad/wrong English, as @WillM suggests, is constantly reproduced by us non-native English speakers just because we see it in some paper and think that therefore it must also be correct.
– Jimmy R.
Nov 29 at 4:50
See also the post Conditional Statements: “only if”
– Mauro ALLEGRANZA
Nov 29 at 7:14
See also the post Conditional Statements: “only if”
– Mauro ALLEGRANZA
Nov 29 at 7:14
|
show 2 more comments
2 Answers
2
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oldest
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The traditional reading of "if $A$, then $B$", i.e. $A to B$, is :
"$A$ is a sufficient condition for $B$" and "$B$ is a necessary condition for $A$".
What happens with :
"$A$ if and only if $B$" ?
It is only a "syntactical" issue. The abbreviation is made of : "$A$ if $B$" and "$A$ only if $B$".
In turn, "$A$ if $B$" is $B to A$, while "$A$ only if $B$" is $A to B$.
Thus, from the point of view of natural language, "$A$ if and only if $B$" is :
"$B to A$ and $A to B$".
But when "if $B$, then $A$", we say that "$A$ is a necessary condition for $B$", and when "if $A$, then $B$" we also say that "$A$ is a sufficient condition for $B$".
Thus, cooking them together, we read :
"$A$ if and only if $B$"
as :
"$A$ is a necessary and sufficient condition for $B$".
answered Nov 29 at 9:59
Mauro ALLEGRANZA
64.1k448111
add a comment |
Indeed, when you use these qualifiers, you always refer to the antecedent: "$A$ is necessary" and "$A$ is sufficient". $A$ is the condition you want to prove, $B$ is the given hypothesis.
Think that they are also used in one-way situations. $x>1$ is sufficient for $x>0$, but it is not necessary.
answered Nov 29 at 10:13
Yves Daoust
124k671221
add a comment |
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2 Answers
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The traditional reading of "if $A$, then $B$", i.e. $A to B$, is :
"$A$ is a sufficient condition for $B$" and "$B$ is a necessary condition for $A$".
What happens with :
"$A$ if and only if $B$" ?
It is only a "syntactical" issue. The abbreviation is made of : "$A$ if $B$" and "$A$ only if $B$".
In turn, "$A$ if $B$" is $B to A$, while "$A$ only if $B$" is $A to B$.
Thus, from the point of view of natural language, "$A$ if and only if $B$" is :
"$B to A$ and $A to B$".
But when "if $B$, then $A$", we say that "$A$ is a necessary condition for $B$", and when "if $A$, then $B$" we also say that "$A$ is a sufficient condition for $B$".
Thus, cooking them together, we read :
"$A$ if and only if $B$"
as :
"$A$ is a necessary and sufficient condition for $B$".
answered Nov 29 at 9:59
Mauro ALLEGRANZA
64.1k448111
add a comment |
The traditional reading of "if $A$, then $B$", i.e. $A to B$, is :
"$A$ is a sufficient condition for $B$" and "$B$ is a necessary condition for $A$".
What happens with :
"$A$ if and only if $B$" ?
It is only a "syntactical" issue. The abbreviation is made of : "$A$ if $B$" and "$A$ only if $B$".
In turn, "$A$ if $B$" is $B to A$, while "$A$ only if $B$" is $A to B$.
Thus, from the point of view of natural language, "$A$ if and only if $B$" is :
"$B to A$ and $A to B$".
But when "if $B$, then $A$", we say that "$A$ is a necessary condition for $B$", and when "if $A$, then $B$" we also say that "$A$ is a sufficient condition for $B$".
Thus, cooking them together, we read :
"$A$ if and only if $B$"
as :
"$A$ is a necessary and sufficient condition for $B$".
answered Nov 29 at 9:59
Mauro ALLEGRANZA
64.1k448111
add a comment |
The traditional reading of "if $A$, then $B$", i.e. $A to B$, is :
"$A$ is a sufficient condition for $B$" and "$B$ is a necessary condition for $A$".
What happens with :
"$A$ if and only if $B$" ?
It is only a "syntactical" issue. The abbreviation is made of : "$A$ if $B$" and "$A$ only if $B$".
In turn, "$A$ if $B$" is $B to A$, while "$A$ only if $B$" is $A to B$.
Thus, from the point of view of natural language, "$A$ if and only if $B$" is :
"$B to A$ and $A to B$".
But when "if $B$, then $A$", we say that "$A$ is a necessary condition for $B$", and when "if $A$, then $B$" we also say that "$A$ is a sufficient condition for $B$".
Thus, cooking them together, we read :
"$A$ if and only if $B$"
as :
"$A$ is a necessary and sufficient condition for $B$".
answered Nov 29 at 9:59
Mauro ALLEGRANZA
64.1k448111
The traditional reading of "if $A$, then $B$", i.e. $A to B$, is :
"$A$ is a sufficient condition for $B$" and "$B$ is a necessary condition for $A$".
What happens with :
"$A$ if and only if $B$" ?
It is only a "syntactical" issue. The abbreviation is made of : "$A$ if $B$" and "$A$ only if $B$".
In turn, "$A$ if $B$" is $B to A$, while "$A$ only if $B$" is $A to B$.
Thus, from the point of view of natural language, "$A$ if and only if $B$" is :
"$B to A$ and $A to B$".
But when "if $B$, then $A$", we say that "$A$ is a necessary condition for $B$", and when "if $A$, then $B$" we also say that "$A$ is a sufficient condition for $B$".
Thus, cooking them together, we read :
"$A$ if and only if $B$"
as :
"$A$ is a necessary and sufficient condition for $B$".
answered Nov 29 at 9:59
Mauro ALLEGRANZA
64.1k448111
edited Nov 29 at 10:08
answered Nov 29 at 9:59
Mauro ALLEGRANZA
64.1k448111
answered Nov 29 at 9:59
Mauro ALLEGRANZA
64.1k448111
answered Nov 29 at 9:59
Mauro ALLEGRANZA
64.1k448111
64.1k448111
add a comment |
add a comment |
Indeed, when you use these qualifiers, you always refer to the antecedent: "$A$ is necessary" and "$A$ is sufficient". $A$ is the condition you want to prove, $B$ is the given hypothesis.
Think that they are also used in one-way situations. $x>1$ is sufficient for $x>0$, but it is not necessary.
answered Nov 29 at 10:13
Yves Daoust
124k671221
add a comment |
Indeed, when you use these qualifiers, you always refer to the antecedent: "$A$ is necessary" and "$A$ is sufficient". $A$ is the condition you want to prove, $B$ is the given hypothesis.
Think that they are also used in one-way situations. $x>1$ is sufficient for $x>0$, but it is not necessary.
answered Nov 29 at 10:13
Yves Daoust
124k671221
add a comment |
Indeed, when you use these qualifiers, you always refer to the antecedent: "$A$ is necessary" and "$A$ is sufficient". $A$ is the condition you want to prove, $B$ is the given hypothesis.
Think that they are also used in one-way situations. $x>1$ is sufficient for $x>0$, but it is not necessary.
answered Nov 29 at 10:13
Yves Daoust
124k671221
Indeed, when you use these qualifiers, you always refer to the antecedent: "$A$ is necessary" and "$A$ is sufficient". $A$ is the condition you want to prove, $B$ is the given hypothesis.
Think that they are also used in one-way situations. $x>1$ is sufficient for $x>0$, but it is not necessary.
answered Nov 29 at 10:13
Yves Daoust
124k671221
edited Nov 29 at 10:19
answered Nov 29 at 10:13
Yves Daoust
124k671221
answered Nov 29 at 10:13
Yves Daoust
124k671221
answered Nov 29 at 10:13
Yves Daoust
124k671221
124k671221
add a comment |
add a comment |
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It is often the case (I do because it is better English) that authors DO write "For A to be the case, it is necessary and sufficient, that B should hold as well." So, I would never ever write "if and only if" or the uglier version "iff."
– Will M.
Nov 29 at 4:43
1
In my experience, authors always keep the ordering of $A$ and $B$. That is, proof of sufficiency means $A implies B$ while proof of necessity means $A impliedby B$. The arbitrariness is only there when you switch the placement of $A$ and $B$.
– Ron
Nov 29 at 4:48
1
I am confused. In the highlighted part you say that the (Necessity) proof is the proof of $A Rightarrow B$, but below that, you say that that is the (Sufficiency) proof?
– Bram28
Nov 29 at 4:49
I agree, I have encountered this specific one and has left me confused as well. I don't think that there is a consistent pattern and even bad/wrong English, as @WillM suggests, is constantly reproduced by us non-native English speakers just because we see it in some paper and think that therefore it must also be correct.
– Jimmy R.
Nov 29 at 4:50
See also the post Conditional Statements: “only if”
– Mauro ALLEGRANZA
Nov 29 at 7:14