Mixing
How do I write minimal negations for the following statements?
$begingroup$
Write minimal negations for the following statements. (In other words, if the statement below is $P,$ what would $¬P$ mean?)
a) Everybody was kung-fu fighting for at least 10 hours.
b) There is a river with at least two tributaries.
c) No baboons wear bowler hats.
I get somewhat confused when it comes to negating the statement. I haven't fully grasped the concept. but this is what I was thinking and kinda feel like its wrong.
A) There is somebody who was kung-fu fighting for not 10 hours
B) There is not a river with at least two tributaries
C) All baboons wear bowler hats
discrete-mathematics logic
edited Jan 24 at 12:40
Namaste
1
asked Jan 24 at 2:53
Brad GuzmanBrad Guzman
63
$endgroup$
add a comment |
$begingroup$
Write minimal negations for the following statements. (In other words, if the statement below is $P,$ what would $¬P$ mean?)
a) Everybody was kung-fu fighting for at least 10 hours.
b) There is a river with at least two tributaries.
c) No baboons wear bowler hats.
I get somewhat confused when it comes to negating the statement. I haven't fully grasped the concept. but this is what I was thinking and kinda feel like its wrong.
A) There is somebody who was kung-fu fighting for not 10 hours
B) There is not a river with at least two tributaries
C) All baboons wear bowler hats
discrete-mathematics logic
edited Jan 24 at 12:40
Namaste
1
asked Jan 24 at 2:53
Brad GuzmanBrad Guzman
63
$endgroup$
add a comment |
$begingroup$
Write minimal negations for the following statements. (In other words, if the statement below is $P,$ what would $¬P$ mean?)
a) Everybody was kung-fu fighting for at least 10 hours.
b) There is a river with at least two tributaries.
c) No baboons wear bowler hats.
I get somewhat confused when it comes to negating the statement. I haven't fully grasped the concept. but this is what I was thinking and kinda feel like its wrong.
A) There is somebody who was kung-fu fighting for not 10 hours
B) There is not a river with at least two tributaries
C) All baboons wear bowler hats
discrete-mathematics logic
edited Jan 24 at 12:40
Namaste
1
asked Jan 24 at 2:53
Brad GuzmanBrad Guzman
63
$endgroup$
Write minimal negations for the following statements. (In other words, if the statement below is $P,$ what would $¬P$ mean?)
a) Everybody was kung-fu fighting for at least 10 hours.
b) There is a river with at least two tributaries.
c) No baboons wear bowler hats.
I get somewhat confused when it comes to negating the statement. I haven't fully grasped the concept. but this is what I was thinking and kinda feel like its wrong.
A) There is somebody who was kung-fu fighting for not 10 hours
B) There is not a river with at least two tributaries
C) All baboons wear bowler hats
discrete-mathematics logic
discrete-mathematics logic
edited Jan 24 at 12:40
Namaste
1
asked Jan 24 at 2:53
Brad GuzmanBrad Guzman
63
edited Jan 24 at 12:40
Namaste
1
asked Jan 24 at 2:53
Brad GuzmanBrad Guzman
63
edited Jan 24 at 12:40
Namaste
1
edited Jan 24 at 12:40
Namaste
1
edited Jan 24 at 12:40
Namaste
1
1
asked Jan 24 at 2:53
Brad GuzmanBrad Guzman
63
asked Jan 24 at 2:53
Brad GuzmanBrad Guzman
63
asked Jan 24 at 2:53
Brad GuzmanBrad Guzman
63
63
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Your reasoning is not quite correct. As it happens all of your answers are unsatisfactory in some way. Let's go through each:
A) There is somebody who was kung-fu fighting for not 10 hours
This is almost correct. If “$neg$(Everybody is doing X)” then there has to be somebody who is not doing X. However, by saying “not 10 hours” you are mistaken: the correct negation is “less than 10 hours”.
B) There is not a river with at least two tributaries
Whilst this is a technically correct negation, it has not been simplified at all. You're likely expected to produce a sentence of the form “All rivers (...)”. I'll leave you to work this one out yourself; you can probably do so, maybe after reading the tips later on in this answer.
C) All baboons wear bowler hats
This is not a correct negation. To see why, try to negate the false statement “No cats are ginger”. By your logic, the negation, a true statement, should be “All cats are ginger”—a clearly false statement! The correct negation is “There is a baboon that wears a bowler hat”.
The general rules for solving this sort of problem can be written as so:
$neg(text{All $X$s have property $Y$}) Leftrightarrow (text{There is an $X$ without property $Y$})$.- $neg(text{There is an $X$ with property $Y$}) Leftrightarrow (text{All $X$s do not have property $Y$})$
Clearly, a) falls under rule 1, whereas b) falls under rule 2. It may be hard to see, but c) actually also falls under rule 1—can you see why? This should help you understand this kind of statement better.
If you'd like more justification, I recommend reading up on logical quantifiers. If these problems are part of a book or course on logic, you will likely encounter these very soon. Logical quantifiers allow rule 1, for instance, to be written as $neg(forall x, P(x)) Leftrightarrow exists x, neg P(x)$.
answered Jan 24 at 3:16
AJFarmarAJFarmar
15910
$endgroup$
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3085391%2fhow-do-i-write-minimal-negations-for-the-following-statements%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Your reasoning is not quite correct. As it happens all of your answers are unsatisfactory in some way. Let's go through each:
A) There is somebody who was kung-fu fighting for not 10 hours
This is almost correct. If “$neg$(Everybody is doing X)” then there has to be somebody who is not doing X. However, by saying “not 10 hours” you are mistaken: the correct negation is “less than 10 hours”.
B) There is not a river with at least two tributaries
Whilst this is a technically correct negation, it has not been simplified at all. You're likely expected to produce a sentence of the form “All rivers (...)”. I'll leave you to work this one out yourself; you can probably do so, maybe after reading the tips later on in this answer.
C) All baboons wear bowler hats
This is not a correct negation. To see why, try to negate the false statement “No cats are ginger”. By your logic, the negation, a true statement, should be “All cats are ginger”—a clearly false statement! The correct negation is “There is a baboon that wears a bowler hat”.
The general rules for solving this sort of problem can be written as so:
$neg(text{All $X$s have property $Y$}) Leftrightarrow (text{There is an $X$ without property $Y$})$.- $neg(text{There is an $X$ with property $Y$}) Leftrightarrow (text{All $X$s do not have property $Y$})$
Clearly, a) falls under rule 1, whereas b) falls under rule 2. It may be hard to see, but c) actually also falls under rule 1—can you see why? This should help you understand this kind of statement better.
If you'd like more justification, I recommend reading up on logical quantifiers. If these problems are part of a book or course on logic, you will likely encounter these very soon. Logical quantifiers allow rule 1, for instance, to be written as $neg(forall x, P(x)) Leftrightarrow exists x, neg P(x)$.
answered Jan 24 at 3:16
AJFarmarAJFarmar
15910
$endgroup$
add a comment |
$begingroup$
Your reasoning is not quite correct. As it happens all of your answers are unsatisfactory in some way. Let's go through each:
A) There is somebody who was kung-fu fighting for not 10 hours
This is almost correct. If “$neg$(Everybody is doing X)” then there has to be somebody who is not doing X. However, by saying “not 10 hours” you are mistaken: the correct negation is “less than 10 hours”.
B) There is not a river with at least two tributaries
Whilst this is a technically correct negation, it has not been simplified at all. You're likely expected to produce a sentence of the form “All rivers (...)”. I'll leave you to work this one out yourself; you can probably do so, maybe after reading the tips later on in this answer.
C) All baboons wear bowler hats
This is not a correct negation. To see why, try to negate the false statement “No cats are ginger”. By your logic, the negation, a true statement, should be “All cats are ginger”—a clearly false statement! The correct negation is “There is a baboon that wears a bowler hat”.
The general rules for solving this sort of problem can be written as so:
$neg(text{All $X$s have property $Y$}) Leftrightarrow (text{There is an $X$ without property $Y$})$.- $neg(text{There is an $X$ with property $Y$}) Leftrightarrow (text{All $X$s do not have property $Y$})$
Clearly, a) falls under rule 1, whereas b) falls under rule 2. It may be hard to see, but c) actually also falls under rule 1—can you see why? This should help you understand this kind of statement better.
If you'd like more justification, I recommend reading up on logical quantifiers. If these problems are part of a book or course on logic, you will likely encounter these very soon. Logical quantifiers allow rule 1, for instance, to be written as $neg(forall x, P(x)) Leftrightarrow exists x, neg P(x)$.
answered Jan 24 at 3:16
AJFarmarAJFarmar
15910
$endgroup$
add a comment |
$begingroup$
Your reasoning is not quite correct. As it happens all of your answers are unsatisfactory in some way. Let's go through each:
A) There is somebody who was kung-fu fighting for not 10 hours
This is almost correct. If “$neg$(Everybody is doing X)” then there has to be somebody who is not doing X. However, by saying “not 10 hours” you are mistaken: the correct negation is “less than 10 hours”.
B) There is not a river with at least two tributaries
Whilst this is a technically correct negation, it has not been simplified at all. You're likely expected to produce a sentence of the form “All rivers (...)”. I'll leave you to work this one out yourself; you can probably do so, maybe after reading the tips later on in this answer.
C) All baboons wear bowler hats
This is not a correct negation. To see why, try to negate the false statement “No cats are ginger”. By your logic, the negation, a true statement, should be “All cats are ginger”—a clearly false statement! The correct negation is “There is a baboon that wears a bowler hat”.
The general rules for solving this sort of problem can be written as so:
$neg(text{All $X$s have property $Y$}) Leftrightarrow (text{There is an $X$ without property $Y$})$.- $neg(text{There is an $X$ with property $Y$}) Leftrightarrow (text{All $X$s do not have property $Y$})$
Clearly, a) falls under rule 1, whereas b) falls under rule 2. It may be hard to see, but c) actually also falls under rule 1—can you see why? This should help you understand this kind of statement better.
If you'd like more justification, I recommend reading up on logical quantifiers. If these problems are part of a book or course on logic, you will likely encounter these very soon. Logical quantifiers allow rule 1, for instance, to be written as $neg(forall x, P(x)) Leftrightarrow exists x, neg P(x)$.
answered Jan 24 at 3:16
AJFarmarAJFarmar
15910
$endgroup$
Your reasoning is not quite correct. As it happens all of your answers are unsatisfactory in some way. Let's go through each:
A) There is somebody who was kung-fu fighting for not 10 hours
This is almost correct. If “$neg$(Everybody is doing X)” then there has to be somebody who is not doing X. However, by saying “not 10 hours” you are mistaken: the correct negation is “less than 10 hours”.
B) There is not a river with at least two tributaries
Whilst this is a technically correct negation, it has not been simplified at all. You're likely expected to produce a sentence of the form “All rivers (...)”. I'll leave you to work this one out yourself; you can probably do so, maybe after reading the tips later on in this answer.
C) All baboons wear bowler hats
This is not a correct negation. To see why, try to negate the false statement “No cats are ginger”. By your logic, the negation, a true statement, should be “All cats are ginger”—a clearly false statement! The correct negation is “There is a baboon that wears a bowler hat”.
The general rules for solving this sort of problem can be written as so:
$neg(text{All $X$s have property $Y$}) Leftrightarrow (text{There is an $X$ without property $Y$})$.- $neg(text{There is an $X$ with property $Y$}) Leftrightarrow (text{All $X$s do not have property $Y$})$
Clearly, a) falls under rule 1, whereas b) falls under rule 2. It may be hard to see, but c) actually also falls under rule 1—can you see why? This should help you understand this kind of statement better.
If you'd like more justification, I recommend reading up on logical quantifiers. If these problems are part of a book or course on logic, you will likely encounter these very soon. Logical quantifiers allow rule 1, for instance, to be written as $neg(forall x, P(x)) Leftrightarrow exists x, neg P(x)$.
answered Jan 24 at 3:16
AJFarmarAJFarmar
15910
edited Jan 24 at 3:21
answered Jan 24 at 3:16
AJFarmarAJFarmar
15910
answered Jan 24 at 3:16
AJFarmarAJFarmar
15910
answered Jan 24 at 3:16
AJFarmarAJFarmar
15910
15910
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3085391%2fhow-do-i-write-minimal-negations-for-the-following-statements%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
