How do I determine whether a function/map T is one-to-one and/or onto?
$begingroup$
Given $T(x,y,z) = (xy, yz, xz)$, determine whether $T$ is one to one and/or onto.
So far, I have come up with a contradiction that proves $T$ is not one to one:
$T(-1, 1, -1) = (-1, -1, 1)$
and, $T(1, -1, 1) = (-1, -1, 1)$. However, I can't figure out whether or not $T$ is onto... I think it is, but I don't know the reaosn why or how to prove it.
Can I say:
Taking arbitrary elements a,b,c $in mathbb R$, $T(a,b,c) = (ab, bc, ac)$, and therefore any element $x,y,z in mathbb R$ can be generated by $x=ab$, $y = bc$, and $z=ac$?
calculus functions linear-transformations
$endgroup$
add a comment |
$begingroup$
Given $T(x,y,z) = (xy, yz, xz)$, determine whether $T$ is one to one and/or onto.
So far, I have come up with a contradiction that proves $T$ is not one to one:
$T(-1, 1, -1) = (-1, -1, 1)$
and, $T(1, -1, 1) = (-1, -1, 1)$. However, I can't figure out whether or not $T$ is onto... I think it is, but I don't know the reaosn why or how to prove it.
Can I say:
Taking arbitrary elements a,b,c $in mathbb R$, $T(a,b,c) = (ab, bc, ac)$, and therefore any element $x,y,z in mathbb R$ can be generated by $x=ab$, $y = bc$, and $z=ac$?
calculus functions linear-transformations
$endgroup$
add a comment |
$begingroup$
Given $T(x,y,z) = (xy, yz, xz)$, determine whether $T$ is one to one and/or onto.
So far, I have come up with a contradiction that proves $T$ is not one to one:
$T(-1, 1, -1) = (-1, -1, 1)$
and, $T(1, -1, 1) = (-1, -1, 1)$. However, I can't figure out whether or not $T$ is onto... I think it is, but I don't know the reaosn why or how to prove it.
Can I say:
Taking arbitrary elements a,b,c $in mathbb R$, $T(a,b,c) = (ab, bc, ac)$, and therefore any element $x,y,z in mathbb R$ can be generated by $x=ab$, $y = bc$, and $z=ac$?
calculus functions linear-transformations
$endgroup$
Given $T(x,y,z) = (xy, yz, xz)$, determine whether $T$ is one to one and/or onto.
So far, I have come up with a contradiction that proves $T$ is not one to one:
$T(-1, 1, -1) = (-1, -1, 1)$
and, $T(1, -1, 1) = (-1, -1, 1)$. However, I can't figure out whether or not $T$ is onto... I think it is, but I don't know the reaosn why or how to prove it.
Can I say:
Taking arbitrary elements a,b,c $in mathbb R$, $T(a,b,c) = (ab, bc, ac)$, and therefore any element $x,y,z in mathbb R$ can be generated by $x=ab$, $y = bc$, and $z=ac$?
calculus functions linear-transformations
calculus functions linear-transformations
asked Dec 7 '18 at 1:21
JaigusJaigus
2218
2218
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Assuming both the domain and codomain are $mathbb{R}^3$.
Suppose on the contrary that it is onto, then $(-1,-1,-1)$ has a preimage.
Suppose $$-1 =ab, -1=bc, -1=ac$$
then we have $a = -frac{1}b, c= -frac{1}{b}$
and we have $ac= frac1{b^2}>0$ which contracts $ac=-1<0$.
$endgroup$
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%2f3029335%2fhow-do-i-determine-whether-a-function-map-t-is-one-to-one-and-or-onto%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
$begingroup$
Assuming both the domain and codomain are $mathbb{R}^3$.
Suppose on the contrary that it is onto, then $(-1,-1,-1)$ has a preimage.
Suppose $$-1 =ab, -1=bc, -1=ac$$
then we have $a = -frac{1}b, c= -frac{1}{b}$
and we have $ac= frac1{b^2}>0$ which contracts $ac=-1<0$.
$endgroup$
add a comment |
$begingroup$
Assuming both the domain and codomain are $mathbb{R}^3$.
Suppose on the contrary that it is onto, then $(-1,-1,-1)$ has a preimage.
Suppose $$-1 =ab, -1=bc, -1=ac$$
then we have $a = -frac{1}b, c= -frac{1}{b}$
and we have $ac= frac1{b^2}>0$ which contracts $ac=-1<0$.
$endgroup$
add a comment |
$begingroup$
Assuming both the domain and codomain are $mathbb{R}^3$.
Suppose on the contrary that it is onto, then $(-1,-1,-1)$ has a preimage.
Suppose $$-1 =ab, -1=bc, -1=ac$$
then we have $a = -frac{1}b, c= -frac{1}{b}$
and we have $ac= frac1{b^2}>0$ which contracts $ac=-1<0$.
$endgroup$
Assuming both the domain and codomain are $mathbb{R}^3$.
Suppose on the contrary that it is onto, then $(-1,-1,-1)$ has a preimage.
Suppose $$-1 =ab, -1=bc, -1=ac$$
then we have $a = -frac{1}b, c= -frac{1}{b}$
and we have $ac= frac1{b^2}>0$ which contracts $ac=-1<0$.
answered Dec 7 '18 at 1:28
Siong Thye GohSiong Thye Goh
100k1466117
100k1466117
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%2f3029335%2fhow-do-i-determine-whether-a-function-map-t-is-one-to-one-and-or-onto%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