Taking a hypothesis and using a list of tautologies to prove a conclusion
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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
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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
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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
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add a comment |
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2 Answers
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2 Answers
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active
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$begingroup$
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 |
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$begingroup$
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