Simplify [p∧ (¬(¬p v q)) ] v (p ∧ q) so that it become p, q, ¬p, or ¬q
$begingroup$
Had a question on a test that asked for us to simplify (using rules of inference) the following proposition: [p∧ (¬(¬p v q)) ] v (p ∧ q)
to p, q, or their negation (¬p, ¬q). Here is what I did:
1) [p∧ (¬(¬p v q)) ] v (p ∧ q)
2) [p∧ (p ∧ ¬q)) ] v (p ∧ q) DeMorgan's Law
3) [(p∧ p) ∧ ¬q] v (p∧ q) Associative law
4) [p ∧ ¬q] v (p∧ q) Idempotent law
Step 4 is where I got stuck. I didn't know what to do afterwards. So I substituted S= (p∧ q) and did distributive law
5) [p ∧ ¬q] v S
6) (S v p) ∧ (S v ¬q) Distributive law
7) [ (p∧q) v p ] ∧ [ (p∧q) v ¬q] Substitute S=(p∧q) back in
8) [ (pvp) ∧ (pvq) ] ∧ [ (¬q v p) ∧ (¬q v q) ] Distributive law
9) [ p ∧ (pvq) ] ∧ [ (¬q v p) ∧ T ] Idempotent law and Domination law
10) p ∧ (¬q v p) Absorption and Identity law
I stopped there. I didn't know how to further simplify step 10 into p, q, ¬p, or ¬q. I was thinking maybe absorption law? But p ∧ (¬q v p) is not the same as p ∧ (q v p), so I couldn't simplify it to p, correct? Can someone help me further simplify it? Do you think I will get most the marks for this question or was my approach completely wrong?
discrete-mathematics logic propositional-calculus boolean-algebra
edited Feb 16 at 4:01
Bram28
64.7k44793
asked Feb 15 at 22:55
NevNev
466
$endgroup$
add a comment |
$begingroup$
Had a question on a test that asked for us to simplify (using rules of inference) the following proposition: [p∧ (¬(¬p v q)) ] v (p ∧ q)
to p, q, or their negation (¬p, ¬q). Here is what I did:
1) [p∧ (¬(¬p v q)) ] v (p ∧ q)
2) [p∧ (p ∧ ¬q)) ] v (p ∧ q) DeMorgan's Law
3) [(p∧ p) ∧ ¬q] v (p∧ q) Associative law
4) [p ∧ ¬q] v (p∧ q) Idempotent law
Step 4 is where I got stuck. I didn't know what to do afterwards. So I substituted S= (p∧ q) and did distributive law
5) [p ∧ ¬q] v S
6) (S v p) ∧ (S v ¬q) Distributive law
7) [ (p∧q) v p ] ∧ [ (p∧q) v ¬q] Substitute S=(p∧q) back in
8) [ (pvp) ∧ (pvq) ] ∧ [ (¬q v p) ∧ (¬q v q) ] Distributive law
9) [ p ∧ (pvq) ] ∧ [ (¬q v p) ∧ T ] Idempotent law and Domination law
10) p ∧ (¬q v p) Absorption and Identity law
I stopped there. I didn't know how to further simplify step 10 into p, q, ¬p, or ¬q. I was thinking maybe absorption law? But p ∧ (¬q v p) is not the same as p ∧ (q v p), so I couldn't simplify it to p, correct? Can someone help me further simplify it? Do you think I will get most the marks for this question or was my approach completely wrong?
discrete-mathematics logic propositional-calculus boolean-algebra
edited Feb 16 at 4:01
Bram28
64.7k44793
asked Feb 15 at 22:55
NevNev
466
$endgroup$
$begingroup$
By the way, $pland (neg q lor p)$ can be immediately simplified to $p$ by what is known as the absorption law. See for example math.stackexchange.com/questions/2421690/… for some further reading.
$endgroup$
– Minus One-Twelfth
Feb 16 at 1:03
add a comment |
$begingroup$
Had a question on a test that asked for us to simplify (using rules of inference) the following proposition: [p∧ (¬(¬p v q)) ] v (p ∧ q)
to p, q, or their negation (¬p, ¬q). Here is what I did:
1) [p∧ (¬(¬p v q)) ] v (p ∧ q)
2) [p∧ (p ∧ ¬q)) ] v (p ∧ q) DeMorgan's Law
3) [(p∧ p) ∧ ¬q] v (p∧ q) Associative law
4) [p ∧ ¬q] v (p∧ q) Idempotent law
Step 4 is where I got stuck. I didn't know what to do afterwards. So I substituted S= (p∧ q) and did distributive law
5) [p ∧ ¬q] v S
6) (S v p) ∧ (S v ¬q) Distributive law
7) [ (p∧q) v p ] ∧ [ (p∧q) v ¬q] Substitute S=(p∧q) back in
8) [ (pvp) ∧ (pvq) ] ∧ [ (¬q v p) ∧ (¬q v q) ] Distributive law
9) [ p ∧ (pvq) ] ∧ [ (¬q v p) ∧ T ] Idempotent law and Domination law
10) p ∧ (¬q v p) Absorption and Identity law
I stopped there. I didn't know how to further simplify step 10 into p, q, ¬p, or ¬q. I was thinking maybe absorption law? But p ∧ (¬q v p) is not the same as p ∧ (q v p), so I couldn't simplify it to p, correct? Can someone help me further simplify it? Do you think I will get most the marks for this question or was my approach completely wrong?
discrete-mathematics logic propositional-calculus boolean-algebra
edited Feb 16 at 4:01
Bram28
64.7k44793
asked Feb 15 at 22:55
NevNev
466
$endgroup$
Had a question on a test that asked for us to simplify (using rules of inference) the following proposition: [p∧ (¬(¬p v q)) ] v (p ∧ q)
to p, q, or their negation (¬p, ¬q). Here is what I did:
1) [p∧ (¬(¬p v q)) ] v (p ∧ q)
2) [p∧ (p ∧ ¬q)) ] v (p ∧ q) DeMorgan's Law
3) [(p∧ p) ∧ ¬q] v (p∧ q) Associative law
4) [p ∧ ¬q] v (p∧ q) Idempotent law
Step 4 is where I got stuck. I didn't know what to do afterwards. So I substituted S= (p∧ q) and did distributive law
5) [p ∧ ¬q] v S
6) (S v p) ∧ (S v ¬q) Distributive law
7) [ (p∧q) v p ] ∧ [ (p∧q) v ¬q] Substitute S=(p∧q) back in
8) [ (pvp) ∧ (pvq) ] ∧ [ (¬q v p) ∧ (¬q v q) ] Distributive law
9) [ p ∧ (pvq) ] ∧ [ (¬q v p) ∧ T ] Idempotent law and Domination law
10) p ∧ (¬q v p) Absorption and Identity law
I stopped there. I didn't know how to further simplify step 10 into p, q, ¬p, or ¬q. I was thinking maybe absorption law? But p ∧ (¬q v p) is not the same as p ∧ (q v p), so I couldn't simplify it to p, correct? Can someone help me further simplify it? Do you think I will get most the marks for this question or was my approach completely wrong?
discrete-mathematics logic propositional-calculus boolean-algebra
discrete-mathematics logic propositional-calculus boolean-algebra
edited Feb 16 at 4:01
Bram28
64.7k44793
asked Feb 15 at 22:55
NevNev
466
edited Feb 16 at 4:01
Bram28
64.7k44793
asked Feb 15 at 22:55
NevNev
466
edited Feb 16 at 4:01
Bram28
64.7k44793
edited Feb 16 at 4:01
Bram28
64.7k44793
edited Feb 16 at 4:01
Bram28
64.7k44793
64.7k44793
asked Feb 15 at 22:55
NevNev
466
asked Feb 15 at 22:55
NevNev
466
asked Feb 15 at 22:55
NevNev
466
466
$begingroup$
By the way, $pland (neg q lor p)$ can be immediately simplified to $p$ by what is known as the absorption law. See for example math.stackexchange.com/questions/2421690/… for some further reading.
$endgroup$
– Minus One-Twelfth
Feb 16 at 1:03
add a comment |
$begingroup$
By the way, $pland (neg q lor p)$ can be immediately simplified to $p$ by what is known as the absorption law. See for example math.stackexchange.com/questions/2421690/… for some further reading.
$endgroup$
– Minus One-Twelfth
Feb 16 at 1:03
$begingroup$
By the way, $pland (neg q lor p)$ can be immediately simplified to $p$ by what is known as the absorption law. See for example math.stackexchange.com/questions/2421690/… for some further reading.
$endgroup$
– Minus One-Twelfth
Feb 16 at 1:03
$begingroup$
By the way, $pland (neg q lor p)$ can be immediately simplified to $p$ by what is known as the absorption law. See for example math.stackexchange.com/questions/2421690/… for some further reading.
$endgroup$
– Minus One-Twelfth
Feb 16 at 1:03
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
This is a common 'mistake': to Distribute when you should really 'Un-Distribute'. That is, you can see:
$(p land neg q) lor (p land q)$
as the result of applying Distribution to:
$p land (neg q lor q)$
But given that this is an equivalence, that means you can go the other way around as well (i.e. 'un-distribute' ... sometimes I call this 'reverse distribution' or 'factoring')
Note that from your last line we can do this as well:
$p land (neg q lor p) = (bot lor p) land (neg q lor p) = (bot land neg q) lor p = bot lor p =p$
Finally, the fact that $(p land q) lor (p land neg q)$ works out to just $p$ is such a common and handy equivalence that it has been given its own name:
Adjacency
$(p land q) lor (p land neg q) =p$
(and its dual form:) $(p lor q) land (p lor neg q)=p$
So, definitely put that one in your boolean algebra toolbox!
answered Feb 15 at 22:59
Bram28Bram28
64.7k44793
$endgroup$
$begingroup$
Ohhhh you're totally right, that went right over my head. Judging by what I did for this question, do you think I'll lose all the marks?
$endgroup$
– Nev
Feb 15 at 23:03
1
$begingroup$
@nev all your steps were correct, and you did get close, so I would give you a good chunk of the points! :)
$endgroup$
– Bram28
Feb 15 at 23:05
add a comment |
$begingroup$
I'm going to use some shorthand notation: $pq$ for $p land q$, and $p + q$ for $p lor q$. Your proposition is $$p(neg(neg p + q)) + pq.$$ Distributing the negation on the left, we obtain $$p (neg (neg p + q)) + pq equiv pp(neg q) + pq.$$ Note that $pp equiv p$. This now gives us $$p(neg q) + pq.$$
We can "factor out" that $p$ to obtain $$p (neg q) + pq equiv p(neg q + q) equiv p,$$ and we're done.
answered Feb 15 at 23:03
rwboglrwbogl
1,039717
$endgroup$
$begingroup$
Yes I totally see that now, thanks! I really hope I get at least some partial marks. I was supposed to "undo" the distribution rather than distribute again
$endgroup$
– Nev
Feb 15 at 23:04
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
This is a common 'mistake': to Distribute when you should really 'Un-Distribute'. That is, you can see:
$(p land neg q) lor (p land q)$
as the result of applying Distribution to:
$p land (neg q lor q)$
But given that this is an equivalence, that means you can go the other way around as well (i.e. 'un-distribute' ... sometimes I call this 'reverse distribution' or 'factoring')
Note that from your last line we can do this as well:
$p land (neg q lor p) = (bot lor p) land (neg q lor p) = (bot land neg q) lor p = bot lor p =p$
Finally, the fact that $(p land q) lor (p land neg q)$ works out to just $p$ is such a common and handy equivalence that it has been given its own name:
Adjacency
$(p land q) lor (p land neg q) =p$
(and its dual form:) $(p lor q) land (p lor neg q)=p$
So, definitely put that one in your boolean algebra toolbox!
answered Feb 15 at 22:59
Bram28Bram28
64.7k44793
$endgroup$
$begingroup$
Ohhhh you're totally right, that went right over my head. Judging by what I did for this question, do you think I'll lose all the marks?
$endgroup$
– Nev
Feb 15 at 23:03
1
$begingroup$
@nev all your steps were correct, and you did get close, so I would give you a good chunk of the points! :)
$endgroup$
– Bram28
Feb 15 at 23:05
add a comment |
$begingroup$
This is a common 'mistake': to Distribute when you should really 'Un-Distribute'. That is, you can see:
$(p land neg q) lor (p land q)$
as the result of applying Distribution to:
$p land (neg q lor q)$
But given that this is an equivalence, that means you can go the other way around as well (i.e. 'un-distribute' ... sometimes I call this 'reverse distribution' or 'factoring')
Note that from your last line we can do this as well:
$p land (neg q lor p) = (bot lor p) land (neg q lor p) = (bot land neg q) lor p = bot lor p =p$
Finally, the fact that $(p land q) lor (p land neg q)$ works out to just $p$ is such a common and handy equivalence that it has been given its own name:
Adjacency
$(p land q) lor (p land neg q) =p$
(and its dual form:) $(p lor q) land (p lor neg q)=p$
So, definitely put that one in your boolean algebra toolbox!
answered Feb 15 at 22:59
Bram28Bram28
64.7k44793
$endgroup$
$begingroup$
Ohhhh you're totally right, that went right over my head. Judging by what I did for this question, do you think I'll lose all the marks?
$endgroup$
– Nev
Feb 15 at 23:03
1
$begingroup$
@nev all your steps were correct, and you did get close, so I would give you a good chunk of the points! :)
$endgroup$
– Bram28
Feb 15 at 23:05
add a comment |
$begingroup$
This is a common 'mistake': to Distribute when you should really 'Un-Distribute'. That is, you can see:
$(p land neg q) lor (p land q)$
as the result of applying Distribution to:
$p land (neg q lor q)$
But given that this is an equivalence, that means you can go the other way around as well (i.e. 'un-distribute' ... sometimes I call this 'reverse distribution' or 'factoring')
Note that from your last line we can do this as well:
$p land (neg q lor p) = (bot lor p) land (neg q lor p) = (bot land neg q) lor p = bot lor p =p$
Finally, the fact that $(p land q) lor (p land neg q)$ works out to just $p$ is such a common and handy equivalence that it has been given its own name:
Adjacency
$(p land q) lor (p land neg q) =p$
(and its dual form:) $(p lor q) land (p lor neg q)=p$
So, definitely put that one in your boolean algebra toolbox!
answered Feb 15 at 22:59
Bram28Bram28
64.7k44793
$endgroup$
This is a common 'mistake': to Distribute when you should really 'Un-Distribute'. That is, you can see:
$(p land neg q) lor (p land q)$
as the result of applying Distribution to:
$p land (neg q lor q)$
But given that this is an equivalence, that means you can go the other way around as well (i.e. 'un-distribute' ... sometimes I call this 'reverse distribution' or 'factoring')
Note that from your last line we can do this as well:
$p land (neg q lor p) = (bot lor p) land (neg q lor p) = (bot land neg q) lor p = bot lor p =p$
Finally, the fact that $(p land q) lor (p land neg q)$ works out to just $p$ is such a common and handy equivalence that it has been given its own name:
Adjacency
$(p land q) lor (p land neg q) =p$
(and its dual form:) $(p lor q) land (p lor neg q)=p$
So, definitely put that one in your boolean algebra toolbox!
answered Feb 15 at 22:59
Bram28Bram28
64.7k44793
edited Feb 15 at 23:24
answered Feb 15 at 22:59
Bram28Bram28
64.7k44793
answered Feb 15 at 22:59
Bram28Bram28
64.7k44793
answered Feb 15 at 22:59
Bram28Bram28
64.7k44793
64.7k44793
$begingroup$
Ohhhh you're totally right, that went right over my head. Judging by what I did for this question, do you think I'll lose all the marks?
$endgroup$
– Nev
Feb 15 at 23:03
1
$begingroup$
@nev all your steps were correct, and you did get close, so I would give you a good chunk of the points! :)
$endgroup$
– Bram28
Feb 15 at 23:05
add a comment |
$begingroup$
Ohhhh you're totally right, that went right over my head. Judging by what I did for this question, do you think I'll lose all the marks?
$endgroup$
– Nev
Feb 15 at 23:03
1
$begingroup$
@nev all your steps were correct, and you did get close, so I would give you a good chunk of the points! :)
$endgroup$
– Bram28
Feb 15 at 23:05
$begingroup$
Ohhhh you're totally right, that went right over my head. Judging by what I did for this question, do you think I'll lose all the marks?
$endgroup$
– Nev
Feb 15 at 23:03
$begingroup$
Ohhhh you're totally right, that went right over my head. Judging by what I did for this question, do you think I'll lose all the marks?
$endgroup$
– Nev
Feb 15 at 23:03
1
1
$begingroup$
@nev all your steps were correct, and you did get close, so I would give you a good chunk of the points! :)
$endgroup$
– Bram28
Feb 15 at 23:05
$begingroup$
@nev all your steps were correct, and you did get close, so I would give you a good chunk of the points! :)
$endgroup$
– Bram28
Feb 15 at 23:05
add a comment |
$begingroup$
I'm going to use some shorthand notation: $pq$ for $p land q$, and $p + q$ for $p lor q$. Your proposition is $$p(neg(neg p + q)) + pq.$$ Distributing the negation on the left, we obtain $$p (neg (neg p + q)) + pq equiv pp(neg q) + pq.$$ Note that $pp equiv p$. This now gives us $$p(neg q) + pq.$$
We can "factor out" that $p$ to obtain $$p (neg q) + pq equiv p(neg q + q) equiv p,$$ and we're done.
answered Feb 15 at 23:03
rwboglrwbogl
1,039717
$endgroup$
$begingroup$
Yes I totally see that now, thanks! I really hope I get at least some partial marks. I was supposed to "undo" the distribution rather than distribute again
$endgroup$
– Nev
Feb 15 at 23:04
add a comment |
$begingroup$
I'm going to use some shorthand notation: $pq$ for $p land q$, and $p + q$ for $p lor q$. Your proposition is $$p(neg(neg p + q)) + pq.$$ Distributing the negation on the left, we obtain $$p (neg (neg p + q)) + pq equiv pp(neg q) + pq.$$ Note that $pp equiv p$. This now gives us $$p(neg q) + pq.$$
We can "factor out" that $p$ to obtain $$p (neg q) + pq equiv p(neg q + q) equiv p,$$ and we're done.
answered Feb 15 at 23:03
rwboglrwbogl
1,039717
$endgroup$
$begingroup$
Yes I totally see that now, thanks! I really hope I get at least some partial marks. I was supposed to "undo" the distribution rather than distribute again
$endgroup$
– Nev
Feb 15 at 23:04
add a comment |
$begingroup$
I'm going to use some shorthand notation: $pq$ for $p land q$, and $p + q$ for $p lor q$. Your proposition is $$p(neg(neg p + q)) + pq.$$ Distributing the negation on the left, we obtain $$p (neg (neg p + q)) + pq equiv pp(neg q) + pq.$$ Note that $pp equiv p$. This now gives us $$p(neg q) + pq.$$
We can "factor out" that $p$ to obtain $$p (neg q) + pq equiv p(neg q + q) equiv p,$$ and we're done.
answered Feb 15 at 23:03
rwboglrwbogl
1,039717
$endgroup$
I'm going to use some shorthand notation: $pq$ for $p land q$, and $p + q$ for $p lor q$. Your proposition is $$p(neg(neg p + q)) + pq.$$ Distributing the negation on the left, we obtain $$p (neg (neg p + q)) + pq equiv pp(neg q) + pq.$$ Note that $pp equiv p$. This now gives us $$p(neg q) + pq.$$
We can "factor out" that $p$ to obtain $$p (neg q) + pq equiv p(neg q + q) equiv p,$$ and we're done.
answered Feb 15 at 23:03
rwboglrwbogl
1,039717
answered Feb 15 at 23:03
rwboglrwbogl
1,039717
answered Feb 15 at 23:03
rwboglrwbogl
1,039717
answered Feb 15 at 23:03
rwboglrwbogl
1,039717
1,039717
$begingroup$
Yes I totally see that now, thanks! I really hope I get at least some partial marks. I was supposed to "undo" the distribution rather than distribute again
$endgroup$
– Nev
Feb 15 at 23:04
add a comment |
$begingroup$
Yes I totally see that now, thanks! I really hope I get at least some partial marks. I was supposed to "undo" the distribution rather than distribute again
$endgroup$
– Nev
Feb 15 at 23:04
$begingroup$
Yes I totally see that now, thanks! I really hope I get at least some partial marks. I was supposed to "undo" the distribution rather than distribute again
$endgroup$
– Nev
Feb 15 at 23:04
$begingroup$
Yes I totally see that now, thanks! I really hope I get at least some partial marks. I was supposed to "undo" the distribution rather than distribute again
$endgroup$
– Nev
Feb 15 at 23:04
add a comment |
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Required, but never shown
Required, but never shown
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By the way, $pland (neg q lor p)$ can be immediately simplified to $p$ by what is known as the absorption law. See for example math.stackexchange.com/questions/2421690/… for some further reading.
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– Minus One-Twelfth
Feb 16 at 1:03