Why non-increasing is decreasing?
$begingroup$
As far as I know, by definition, non-decreasing means increasing and non-increasing means decreasing. My general question is: why some people use non-increasing and non-decreasing?
In fact, it raises some confusing to me. For example, the sequence
$1,2,3,4$
is increasing and thus the sequence
$4,1,2,3$
is non-increasing. So, based on the definition, it is decreasing, but it is not.
definition
asked Dec 19 '18 at 15:17
zdmzdm
1806
$endgroup$
add a comment |
$begingroup$
As far as I know, by definition, non-decreasing means increasing and non-increasing means decreasing. My general question is: why some people use non-increasing and non-decreasing?
In fact, it raises some confusing to me. For example, the sequence
$1,2,3,4$
is increasing and thus the sequence
$4,1,2,3$
is non-increasing. So, based on the definition, it is decreasing, but it is not.
definition
asked Dec 19 '18 at 15:17
zdmzdm
1806
$endgroup$
3
$begingroup$
What you think of as nonincreasing isn't what I think of as non-increasing.
$endgroup$
– Lord Shark the Unknown
Dec 19 '18 at 15:21
2
$begingroup$
"non-increasing means decreasing" NO : $0,0,0,ldots$ is non-increasing but not decreasing.
$endgroup$
– Mauro ALLEGRANZA
Dec 19 '18 at 15:22
2
$begingroup$
"nonincreasing" $neq$ "not increasing"
$endgroup$
– rschwieb
Dec 19 '18 at 15:24
add a comment |
$begingroup$
As far as I know, by definition, non-decreasing means increasing and non-increasing means decreasing. My general question is: why some people use non-increasing and non-decreasing?
In fact, it raises some confusing to me. For example, the sequence
$1,2,3,4$
is increasing and thus the sequence
$4,1,2,3$
is non-increasing. So, based on the definition, it is decreasing, but it is not.
definition
asked Dec 19 '18 at 15:17
zdmzdm
1806
$endgroup$
As far as I know, by definition, non-decreasing means increasing and non-increasing means decreasing. My general question is: why some people use non-increasing and non-decreasing?
In fact, it raises some confusing to me. For example, the sequence
$1,2,3,4$
is increasing and thus the sequence
$4,1,2,3$
is non-increasing. So, based on the definition, it is decreasing, but it is not.
definition
definition
asked Dec 19 '18 at 15:17
zdmzdm
1806
asked Dec 19 '18 at 15:17
zdmzdm
1806
asked Dec 19 '18 at 15:17
zdmzdm
1806
asked Dec 19 '18 at 15:17
zdmzdm
1806
asked Dec 19 '18 at 15:17
zdmzdm
1806
1806
3
$begingroup$
What you think of as nonincreasing isn't what I think of as non-increasing.
$endgroup$
– Lord Shark the Unknown
Dec 19 '18 at 15:21
2
$begingroup$
"non-increasing means decreasing" NO : $0,0,0,ldots$ is non-increasing but not decreasing.
$endgroup$
– Mauro ALLEGRANZA
Dec 19 '18 at 15:22
2
$begingroup$
"nonincreasing" $neq$ "not increasing"
$endgroup$
– rschwieb
Dec 19 '18 at 15:24
add a comment |
3
$begingroup$
What you think of as nonincreasing isn't what I think of as non-increasing.
$endgroup$
– Lord Shark the Unknown
Dec 19 '18 at 15:21
2
$begingroup$
"non-increasing means decreasing" NO : $0,0,0,ldots$ is non-increasing but not decreasing.
$endgroup$
– Mauro ALLEGRANZA
Dec 19 '18 at 15:22
2
$begingroup$
"nonincreasing" $neq$ "not increasing"
$endgroup$
– rschwieb
Dec 19 '18 at 15:24
3
3
$begingroup$
What you think of as nonincreasing isn't what I think of as non-increasing.
$endgroup$
– Lord Shark the Unknown
Dec 19 '18 at 15:21
$begingroup$
What you think of as nonincreasing isn't what I think of as non-increasing.
$endgroup$
– Lord Shark the Unknown
Dec 19 '18 at 15:21
2
2
$begingroup$
"non-increasing means decreasing" NO : $0,0,0,ldots$ is non-increasing but not decreasing.
$endgroup$
– Mauro ALLEGRANZA
Dec 19 '18 at 15:22
$begingroup$
"non-increasing means decreasing" NO : $0,0,0,ldots$ is non-increasing but not decreasing.
$endgroup$
– Mauro ALLEGRANZA
Dec 19 '18 at 15:22
2
2
$begingroup$
"nonincreasing" $neq$ "not increasing"
$endgroup$
– rschwieb
Dec 19 '18 at 15:24
$begingroup$
"nonincreasing" $neq$ "not increasing"
$endgroup$
– rschwieb
Dec 19 '18 at 15:24
add a comment |
3 Answers
3
active
oldest
votes
$begingroup$
This is a bit of confusing terminology.
"Non-increasing," unfortunately, does not mean "not increasing" - it means "never increasing." So, for example, the sequence $$3,3,2,1$$ is non-increasing - $3not<3, 3not<2, 2not<1$. The definition is similar for non-decreasing. (Note in particular that non-increasing does not imply decreasing, as the above example shows.)
The sequence $$4,1,2,3$$ has both increases ($2$ to $3$) and decreases ($4$ to $1$); so it is - awkwardly - not increasing, not decreasing, not non-increasing, and not non-decreasing.
Ugh!
answered Dec 19 '18 at 15:23
Noah SchweberNoah Schweber
125k10150287
$endgroup$
add a comment |
$begingroup$
My understanding is that a non-increasing sequence is different than a decreasing one in the sense that decreasing one "decreases every time" and non-increasing "doesn't increase at all" So the example you gave wouldn't be non increasing as it increases from 1 to 2 and 2 to 3.
answered Dec 19 '18 at 15:23
OfyaOfya
5198
$endgroup$
add a comment |
$begingroup$
There are three conditions of change to the next event. Increasing, decreasing and remaining constant. Negative to these three states are non-increasing, non-decreasing and not remaining constant, i.e, varying.
So they are not mutually exclusive.
answered Dec 19 '18 at 15:25
NarasimhamNarasimham
20.8k52158
$endgroup$
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%2f3046491%2fwhy-non-increasing-is-decreasing%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
This is a bit of confusing terminology.
"Non-increasing," unfortunately, does not mean "not increasing" - it means "never increasing." So, for example, the sequence $$3,3,2,1$$ is non-increasing - $3not<3, 3not<2, 2not<1$. The definition is similar for non-decreasing. (Note in particular that non-increasing does not imply decreasing, as the above example shows.)
The sequence $$4,1,2,3$$ has both increases ($2$ to $3$) and decreases ($4$ to $1$); so it is - awkwardly - not increasing, not decreasing, not non-increasing, and not non-decreasing.
Ugh!
answered Dec 19 '18 at 15:23
Noah SchweberNoah Schweber
125k10150287
$endgroup$
add a comment |
$begingroup$
This is a bit of confusing terminology.
"Non-increasing," unfortunately, does not mean "not increasing" - it means "never increasing." So, for example, the sequence $$3,3,2,1$$ is non-increasing - $3not<3, 3not<2, 2not<1$. The definition is similar for non-decreasing. (Note in particular that non-increasing does not imply decreasing, as the above example shows.)
The sequence $$4,1,2,3$$ has both increases ($2$ to $3$) and decreases ($4$ to $1$); so it is - awkwardly - not increasing, not decreasing, not non-increasing, and not non-decreasing.
Ugh!
answered Dec 19 '18 at 15:23
Noah SchweberNoah Schweber
125k10150287
$endgroup$
add a comment |
$begingroup$
This is a bit of confusing terminology.
"Non-increasing," unfortunately, does not mean "not increasing" - it means "never increasing." So, for example, the sequence $$3,3,2,1$$ is non-increasing - $3not<3, 3not<2, 2not<1$. The definition is similar for non-decreasing. (Note in particular that non-increasing does not imply decreasing, as the above example shows.)
The sequence $$4,1,2,3$$ has both increases ($2$ to $3$) and decreases ($4$ to $1$); so it is - awkwardly - not increasing, not decreasing, not non-increasing, and not non-decreasing.
Ugh!
answered Dec 19 '18 at 15:23
Noah SchweberNoah Schweber
125k10150287
$endgroup$
This is a bit of confusing terminology.
"Non-increasing," unfortunately, does not mean "not increasing" - it means "never increasing." So, for example, the sequence $$3,3,2,1$$ is non-increasing - $3not<3, 3not<2, 2not<1$. The definition is similar for non-decreasing. (Note in particular that non-increasing does not imply decreasing, as the above example shows.)
The sequence $$4,1,2,3$$ has both increases ($2$ to $3$) and decreases ($4$ to $1$); so it is - awkwardly - not increasing, not decreasing, not non-increasing, and not non-decreasing.
Ugh!
answered Dec 19 '18 at 15:23
Noah SchweberNoah Schweber
125k10150287
answered Dec 19 '18 at 15:23
Noah SchweberNoah Schweber
125k10150287
answered Dec 19 '18 at 15:23
Noah SchweberNoah Schweber
125k10150287
answered Dec 19 '18 at 15:23
Noah SchweberNoah Schweber
125k10150287
125k10150287
add a comment |
add a comment |
$begingroup$
My understanding is that a non-increasing sequence is different than a decreasing one in the sense that decreasing one "decreases every time" and non-increasing "doesn't increase at all" So the example you gave wouldn't be non increasing as it increases from 1 to 2 and 2 to 3.
answered Dec 19 '18 at 15:23
OfyaOfya
5198
$endgroup$
add a comment |
$begingroup$
My understanding is that a non-increasing sequence is different than a decreasing one in the sense that decreasing one "decreases every time" and non-increasing "doesn't increase at all" So the example you gave wouldn't be non increasing as it increases from 1 to 2 and 2 to 3.
answered Dec 19 '18 at 15:23
OfyaOfya
5198
$endgroup$
add a comment |
$begingroup$
My understanding is that a non-increasing sequence is different than a decreasing one in the sense that decreasing one "decreases every time" and non-increasing "doesn't increase at all" So the example you gave wouldn't be non increasing as it increases from 1 to 2 and 2 to 3.
answered Dec 19 '18 at 15:23
OfyaOfya
5198
$endgroup$
My understanding is that a non-increasing sequence is different than a decreasing one in the sense that decreasing one "decreases every time" and non-increasing "doesn't increase at all" So the example you gave wouldn't be non increasing as it increases from 1 to 2 and 2 to 3.
answered Dec 19 '18 at 15:23
OfyaOfya
5198
answered Dec 19 '18 at 15:23
OfyaOfya
5198
answered Dec 19 '18 at 15:23
OfyaOfya
5198
answered Dec 19 '18 at 15:23
OfyaOfya
5198
5198
add a comment |
add a comment |
$begingroup$
There are three conditions of change to the next event. Increasing, decreasing and remaining constant. Negative to these three states are non-increasing, non-decreasing and not remaining constant, i.e, varying.
So they are not mutually exclusive.
answered Dec 19 '18 at 15:25
NarasimhamNarasimham
20.8k52158
$endgroup$
add a comment |
$begingroup$
There are three conditions of change to the next event. Increasing, decreasing and remaining constant. Negative to these three states are non-increasing, non-decreasing and not remaining constant, i.e, varying.
So they are not mutually exclusive.
answered Dec 19 '18 at 15:25
NarasimhamNarasimham
20.8k52158
$endgroup$
add a comment |
$begingroup$
There are three conditions of change to the next event. Increasing, decreasing and remaining constant. Negative to these three states are non-increasing, non-decreasing and not remaining constant, i.e, varying.
So they are not mutually exclusive.
answered Dec 19 '18 at 15:25
NarasimhamNarasimham
20.8k52158
$endgroup$
There are three conditions of change to the next event. Increasing, decreasing and remaining constant. Negative to these three states are non-increasing, non-decreasing and not remaining constant, i.e, varying.
So they are not mutually exclusive.
answered Dec 19 '18 at 15:25
NarasimhamNarasimham
20.8k52158
answered Dec 19 '18 at 15:25
NarasimhamNarasimham
20.8k52158
answered Dec 19 '18 at 15:25
NarasimhamNarasimham
20.8k52158
answered Dec 19 '18 at 15:25
NarasimhamNarasimham
20.8k52158
20.8k52158
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%2f3046491%2fwhy-non-increasing-is-decreasing%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
3
$begingroup$
What you think of as nonincreasing isn't what I think of as non-increasing.
$endgroup$
– Lord Shark the Unknown
Dec 19 '18 at 15:21
2
$begingroup$
"non-increasing means decreasing" NO : $0,0,0,ldots$ is non-increasing but not decreasing.
$endgroup$
– Mauro ALLEGRANZA
Dec 19 '18 at 15:22
2
$begingroup$
"nonincreasing" $neq$ "not increasing"
$endgroup$
– rschwieb
Dec 19 '18 at 15:24