Taking a hypothesis and using a list of tautologies to prove a conclusion
Multi tool use
$begingroup$
So I had this problem on an old homework that I didn't really understand.
In each part a list of hypotheses are given. These hypotheses are assumed to be true. Using tautologies, you are to establish a desired conclusion. Indicate which tautology you are using to justify each step.
Hypothesis: r $Rightarrow$ $lnot$s , $lnot$r $Rightarrow$ $lnot$t , $lnot$t $Rightarrow$ u, v$Rightarrow$s
Conclusion: $lnot$v $lor$ u
So I went to office hours for this question and my professor pretty much reiterated the hint section in the back of hour textbook. What I am stuck on is how to use all these hypotheses to prove this conclusion. Can anybody guide me on how to approach a problem like this?
here is a list of tautologies for reference: http://www.math.ucsd.edu/~jeggers/math109/tautologies.pdf
logic proof-writing induction
asked Jan 17 at 18:54
Zach LedermanZach Lederman
61
$endgroup$
add a comment |
$begingroup$
So I had this problem on an old homework that I didn't really understand.
In each part a list of hypotheses are given. These hypotheses are assumed to be true. Using tautologies, you are to establish a desired conclusion. Indicate which tautology you are using to justify each step.
Hypothesis: r $Rightarrow$ $lnot$s , $lnot$r $Rightarrow$ $lnot$t , $lnot$t $Rightarrow$ u, v$Rightarrow$s
Conclusion: $lnot$v $lor$ u
So I went to office hours for this question and my professor pretty much reiterated the hint section in the back of hour textbook. What I am stuck on is how to use all these hypotheses to prove this conclusion. Can anybody guide me on how to approach a problem like this?
here is a list of tautologies for reference: http://www.math.ucsd.edu/~jeggers/math109/tautologies.pdf
logic proof-writing induction
asked Jan 17 at 18:54
Zach LedermanZach Lederman
61
$endgroup$
$begingroup$
The conclusion $neg vvee u$ is equivalent to $vimplies u$. Starting with the last hypothesis, can you use the chain rule (hypothetical syllogism) to arrive at the conclusion?
$endgroup$
– Shubham Johri
Jan 17 at 19:01
$begingroup$
How exactly are these tautologies to be used in a proof? Indeed, how is a proof defined by your professor? Did you get an example of what it is supposed to look like? Some other problem like this with a solution acceptable to your professor? If so, could you add it to your post?
$endgroup$
– Bram28
Jan 17 at 20:33
add a comment |
$begingroup$
So I had this problem on an old homework that I didn't really understand.
In each part a list of hypotheses are given. These hypotheses are assumed to be true. Using tautologies, you are to establish a desired conclusion. Indicate which tautology you are using to justify each step.
Hypothesis: r $Rightarrow$ $lnot$s , $lnot$r $Rightarrow$ $lnot$t , $lnot$t $Rightarrow$ u, v$Rightarrow$s
Conclusion: $lnot$v $lor$ u
So I went to office hours for this question and my professor pretty much reiterated the hint section in the back of hour textbook. What I am stuck on is how to use all these hypotheses to prove this conclusion. Can anybody guide me on how to approach a problem like this?
here is a list of tautologies for reference: http://www.math.ucsd.edu/~jeggers/math109/tautologies.pdf
logic proof-writing induction
asked Jan 17 at 18:54
Zach LedermanZach Lederman
61
$endgroup$
So I had this problem on an old homework that I didn't really understand.
In each part a list of hypotheses are given. These hypotheses are assumed to be true. Using tautologies, you are to establish a desired conclusion. Indicate which tautology you are using to justify each step.
Hypothesis: r $Rightarrow$ $lnot$s , $lnot$r $Rightarrow$ $lnot$t , $lnot$t $Rightarrow$ u, v$Rightarrow$s
Conclusion: $lnot$v $lor$ u
So I went to office hours for this question and my professor pretty much reiterated the hint section in the back of hour textbook. What I am stuck on is how to use all these hypotheses to prove this conclusion. Can anybody guide me on how to approach a problem like this?
here is a list of tautologies for reference: http://www.math.ucsd.edu/~jeggers/math109/tautologies.pdf
logic proof-writing induction
logic proof-writing induction
asked Jan 17 at 18:54
Zach LedermanZach Lederman
61
asked Jan 17 at 18:54
Zach LedermanZach Lederman
61
asked Jan 17 at 18:54
Zach LedermanZach Lederman
61
asked Jan 17 at 18:54
Zach LedermanZach Lederman
61
asked Jan 17 at 18:54
Zach LedermanZach Lederman
61
61
$begingroup$
The conclusion $neg vvee u$ is equivalent to $vimplies u$. Starting with the last hypothesis, can you use the chain rule (hypothetical syllogism) to arrive at the conclusion?
$endgroup$
– Shubham Johri
Jan 17 at 19:01
$begingroup$
How exactly are these tautologies to be used in a proof? Indeed, how is a proof defined by your professor? Did you get an example of what it is supposed to look like? Some other problem like this with a solution acceptable to your professor? If so, could you add it to your post?
$endgroup$
– Bram28
Jan 17 at 20:33
add a comment |
$begingroup$
The conclusion $neg vvee u$ is equivalent to $vimplies u$. Starting with the last hypothesis, can you use the chain rule (hypothetical syllogism) to arrive at the conclusion?
$endgroup$
– Shubham Johri
Jan 17 at 19:01
$begingroup$
How exactly are these tautologies to be used in a proof? Indeed, how is a proof defined by your professor? Did you get an example of what it is supposed to look like? Some other problem like this with a solution acceptable to your professor? If so, could you add it to your post?
$endgroup$
– Bram28
Jan 17 at 20:33
$begingroup$
The conclusion $neg vvee u$ is equivalent to $vimplies u$. Starting with the last hypothesis, can you use the chain rule (hypothetical syllogism) to arrive at the conclusion?
$endgroup$
– Shubham Johri
Jan 17 at 19:01
$begingroup$
The conclusion $neg vvee u$ is equivalent to $vimplies u$. Starting with the last hypothesis, can you use the chain rule (hypothetical syllogism) to arrive at the conclusion?
$endgroup$
– Shubham Johri
Jan 17 at 19:01
$begingroup$
How exactly are these tautologies to be used in a proof? Indeed, how is a proof defined by your professor? Did you get an example of what it is supposed to look like? Some other problem like this with a solution acceptable to your professor? If so, could you add it to your post?
$endgroup$
– Bram28
Jan 17 at 20:33
$begingroup$
How exactly are these tautologies to be used in a proof? Indeed, how is a proof defined by your professor? Did you get an example of what it is supposed to look like? Some other problem like this with a solution acceptable to your professor? If so, could you add it to your post?
$endgroup$
– Bram28
Jan 17 at 20:33
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
One approach is to see what you feel you can deduce from the given hypotheses, and then separately concentrate on how to express that deduction using tautologies. For example: $r$ implies $lnot s$, and $lnot s$ implies $lnot v$ (by the contrapositive of $vimplies s$; so if $r$ is true then $lnot v$ is true. Similarly, what can you deduce if $lnot r$ is true? And then, one of $r$ and $lnot r$ has to be true....
answered Jan 17 at 18:58
Greg MartinGreg Martin
35.1k23263
$endgroup$
$begingroup$
thank you! This is all very new to me but I am starting to really like proof by induction.
$endgroup$
– Zach Lederman
Jan 17 at 19:35
add a comment |
$begingroup$
begin{cases}vimplies s&(1)text{ Given}\rimpliesneg s&(2)text{ Given}\simpliesneg r&(3)text{ Contrapositive of }(2)\vimpliesneg r&(4)text{ Hypothetical Syllogism }(1),(3)\neg rimpliesneg t&(5)text{ Given}\vimpliesneg t&(6)text{ Hypothetical Syllogism }(4),(5)\neg timplies u&(7)text{ Given}\vimplies u&(8)text{ Hypothetical Syllogism }(6),(7)\neg vvee u&text{Implication }(8)end{cases}
answered Jan 17 at 19:15
Shubham JohriShubham Johri
5,172717
$endgroup$
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%2f3077370%2ftaking-a-hypothesis-and-using-a-list-of-tautologies-to-prove-a-conclusion%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
One approach is to see what you feel you can deduce from the given hypotheses, and then separately concentrate on how to express that deduction using tautologies. For example: $r$ implies $lnot s$, and $lnot s$ implies $lnot v$ (by the contrapositive of $vimplies s$; so if $r$ is true then $lnot v$ is true. Similarly, what can you deduce if $lnot r$ is true? And then, one of $r$ and $lnot r$ has to be true....
answered Jan 17 at 18:58
Greg MartinGreg Martin
35.1k23263
$endgroup$
$begingroup$
thank you! This is all very new to me but I am starting to really like proof by induction.
$endgroup$
– Zach Lederman
Jan 17 at 19:35
add a comment |
$begingroup$
One approach is to see what you feel you can deduce from the given hypotheses, and then separately concentrate on how to express that deduction using tautologies. For example: $r$ implies $lnot s$, and $lnot s$ implies $lnot v$ (by the contrapositive of $vimplies s$; so if $r$ is true then $lnot v$ is true. Similarly, what can you deduce if $lnot r$ is true? And then, one of $r$ and $lnot r$ has to be true....
answered Jan 17 at 18:58
Greg MartinGreg Martin
35.1k23263
$endgroup$
$begingroup$
thank you! This is all very new to me but I am starting to really like proof by induction.
$endgroup$
– Zach Lederman
Jan 17 at 19:35
add a comment |
$begingroup$
One approach is to see what you feel you can deduce from the given hypotheses, and then separately concentrate on how to express that deduction using tautologies. For example: $r$ implies $lnot s$, and $lnot s$ implies $lnot v$ (by the contrapositive of $vimplies s$; so if $r$ is true then $lnot v$ is true. Similarly, what can you deduce if $lnot r$ is true? And then, one of $r$ and $lnot r$ has to be true....
answered Jan 17 at 18:58
Greg MartinGreg Martin
35.1k23263
$endgroup$
One approach is to see what you feel you can deduce from the given hypotheses, and then separately concentrate on how to express that deduction using tautologies. For example: $r$ implies $lnot s$, and $lnot s$ implies $lnot v$ (by the contrapositive of $vimplies s$; so if $r$ is true then $lnot v$ is true. Similarly, what can you deduce if $lnot r$ is true? And then, one of $r$ and $lnot r$ has to be true....
answered Jan 17 at 18:58
Greg MartinGreg Martin
35.1k23263
answered Jan 17 at 18:58
Greg MartinGreg Martin
35.1k23263
answered Jan 17 at 18:58
Greg MartinGreg Martin
35.1k23263
answered Jan 17 at 18:58
Greg MartinGreg Martin
35.1k23263
35.1k23263
$begingroup$
thank you! This is all very new to me but I am starting to really like proof by induction.
$endgroup$
– Zach Lederman
Jan 17 at 19:35
add a comment |
$begingroup$
thank you! This is all very new to me but I am starting to really like proof by induction.
$endgroup$
– Zach Lederman
Jan 17 at 19:35
$begingroup$
thank you! This is all very new to me but I am starting to really like proof by induction.
$endgroup$
– Zach Lederman
Jan 17 at 19:35
$begingroup$
thank you! This is all very new to me but I am starting to really like proof by induction.
$endgroup$
– Zach Lederman
Jan 17 at 19:35
add a comment |
$begingroup$
begin{cases}vimplies s&(1)text{ Given}\rimpliesneg s&(2)text{ Given}\simpliesneg r&(3)text{ Contrapositive of }(2)\vimpliesneg r&(4)text{ Hypothetical Syllogism }(1),(3)\neg rimpliesneg t&(5)text{ Given}\vimpliesneg t&(6)text{ Hypothetical Syllogism }(4),(5)\neg timplies u&(7)text{ Given}\vimplies u&(8)text{ Hypothetical Syllogism }(6),(7)\neg vvee u&text{Implication }(8)end{cases}
answered Jan 17 at 19:15
Shubham JohriShubham Johri
5,172717
$endgroup$
add a comment |
$begingroup$
begin{cases}vimplies s&(1)text{ Given}\rimpliesneg s&(2)text{ Given}\simpliesneg r&(3)text{ Contrapositive of }(2)\vimpliesneg r&(4)text{ Hypothetical Syllogism }(1),(3)\neg rimpliesneg t&(5)text{ Given}\vimpliesneg t&(6)text{ Hypothetical Syllogism }(4),(5)\neg timplies u&(7)text{ Given}\vimplies u&(8)text{ Hypothetical Syllogism }(6),(7)\neg vvee u&text{Implication }(8)end{cases}
answered Jan 17 at 19:15
Shubham JohriShubham Johri
5,172717
$endgroup$
add a comment |
$begingroup$
begin{cases}vimplies s&(1)text{ Given}\rimpliesneg s&(2)text{ Given}\simpliesneg r&(3)text{ Contrapositive of }(2)\vimpliesneg r&(4)text{ Hypothetical Syllogism }(1),(3)\neg rimpliesneg t&(5)text{ Given}\vimpliesneg t&(6)text{ Hypothetical Syllogism }(4),(5)\neg timplies u&(7)text{ Given}\vimplies u&(8)text{ Hypothetical Syllogism }(6),(7)\neg vvee u&text{Implication }(8)end{cases}
answered Jan 17 at 19:15
Shubham JohriShubham Johri
5,172717
$endgroup$
begin{cases}vimplies s&(1)text{ Given}\rimpliesneg s&(2)text{ Given}\simpliesneg r&(3)text{ Contrapositive of }(2)\vimpliesneg r&(4)text{ Hypothetical Syllogism }(1),(3)\neg rimpliesneg t&(5)text{ Given}\vimpliesneg t&(6)text{ Hypothetical Syllogism }(4),(5)\neg timplies u&(7)text{ Given}\vimplies u&(8)text{ Hypothetical Syllogism }(6),(7)\neg vvee u&text{Implication }(8)end{cases}
answered Jan 17 at 19:15
Shubham JohriShubham Johri
5,172717
answered Jan 17 at 19:15
Shubham JohriShubham Johri
5,172717
answered Jan 17 at 19:15
Shubham JohriShubham Johri
5,172717
answered Jan 17 at 19:15
Shubham JohriShubham Johri
5,172717
5,172717
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%2f3077370%2ftaking-a-hypothesis-and-using-a-list-of-tautologies-to-prove-a-conclusion%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
5cfNCOYuDXzlrtmNE8,AGCLbY2q,l9hjw6SThCCT21s,EpsH
$begingroup$
The conclusion $neg vvee u$ is equivalent to $vimplies u$. Starting with the last hypothesis, can you use the chain rule (hypothetical syllogism) to arrive at the conclusion?
$endgroup$
– Shubham Johri
Jan 17 at 19:01
$begingroup$
How exactly are these tautologies to be used in a proof? Indeed, how is a proof defined by your professor? Did you get an example of what it is supposed to look like? Some other problem like this with a solution acceptable to your professor? If so, could you add it to your post?
$endgroup$
– Bram28
Jan 17 at 20:33