Two different Venn diagrams for $p rightarrow q$ implication (subset diagram and $thicksim p space text{or}...
$begingroup$
I might be confused about something fundamental here.
Why is this single notion of implication generating two very different Venn diagrams?
One is diagram of a subset relationship, which makes sense. $Prightarrow Q$ means every member of $P$ is a member of $Q$. So on the diagram it would be $P$ inside $Q.$
Second is the diagram of the logical equivalent to implication: $thicksim P space text{or} space Q,$ which looks obviously very different.
How do I reconcile the two different Venn diagrams?
logic
edited Dec 31 '18 at 1:12
Gaby Alfonso
1,028317
asked Dec 30 '18 at 23:48
aNameLikeAnyOtheraNameLikeAnyOther
32
$endgroup$
add a comment |
$begingroup$
I might be confused about something fundamental here.
Why is this single notion of implication generating two very different Venn diagrams?
One is diagram of a subset relationship, which makes sense. $Prightarrow Q$ means every member of $P$ is a member of $Q$. So on the diagram it would be $P$ inside $Q.$
Second is the diagram of the logical equivalent to implication: $thicksim P space text{or} space Q,$ which looks obviously very different.
How do I reconcile the two different Venn diagrams?
logic
edited Dec 31 '18 at 1:12
Gaby Alfonso
1,028317
asked Dec 30 '18 at 23:48
aNameLikeAnyOtheraNameLikeAnyOther
32
$endgroup$
$begingroup$
The second Venn should have some region unshaded. If that part is shrunk to nothing, result should look like first Venn.
$endgroup$
– coffeemath
Dec 31 '18 at 0:05
$begingroup$
But what about the rest of the universe in the second Venn? That still looks different from the first one
$endgroup$
– aNameLikeAnyOther
Dec 31 '18 at 0:12
$begingroup$
What is 'the rest of the universe'? You seem to be using two different ways of thinking about things inconsistently. First expressing the implication relationship by placing one circle inside the other, and then expressing the other relationship by shading. Try expressing the implication in terms of shading. (Also remember that $Pto Q$ holds when $P$ is false).
$endgroup$
– spaceisdarkgreen
Dec 31 '18 at 0:38
$begingroup$
By the rest of the universe I meant all the shaded area outside both the P and Q (in the second Venn, (~P or Q) one.) So there are two kinds of Venn diagrams: with and without shading? The one without shading is used express subset relationship, and the one with shading is for demonstrating boolean functions?
$endgroup$
– aNameLikeAnyOther
Dec 31 '18 at 2:17
add a comment |
$begingroup$
I might be confused about something fundamental here.
Why is this single notion of implication generating two very different Venn diagrams?
One is diagram of a subset relationship, which makes sense. $Prightarrow Q$ means every member of $P$ is a member of $Q$. So on the diagram it would be $P$ inside $Q.$
Second is the diagram of the logical equivalent to implication: $thicksim P space text{or} space Q,$ which looks obviously very different.
How do I reconcile the two different Venn diagrams?
logic
edited Dec 31 '18 at 1:12
Gaby Alfonso
1,028317
asked Dec 30 '18 at 23:48
aNameLikeAnyOtheraNameLikeAnyOther
32
$endgroup$
I might be confused about something fundamental here.
Why is this single notion of implication generating two very different Venn diagrams?
One is diagram of a subset relationship, which makes sense. $Prightarrow Q$ means every member of $P$ is a member of $Q$. So on the diagram it would be $P$ inside $Q.$
Second is the diagram of the logical equivalent to implication: $thicksim P space text{or} space Q,$ which looks obviously very different.
How do I reconcile the two different Venn diagrams?
logic
logic
edited Dec 31 '18 at 1:12
Gaby Alfonso
1,028317
asked Dec 30 '18 at 23:48
aNameLikeAnyOtheraNameLikeAnyOther
32
edited Dec 31 '18 at 1:12
Gaby Alfonso
1,028317
asked Dec 30 '18 at 23:48
aNameLikeAnyOtheraNameLikeAnyOther
32
edited Dec 31 '18 at 1:12
Gaby Alfonso
1,028317
edited Dec 31 '18 at 1:12
Gaby Alfonso
1,028317
edited Dec 31 '18 at 1:12
Gaby Alfonso
1,028317
1,028317
asked Dec 30 '18 at 23:48
aNameLikeAnyOtheraNameLikeAnyOther
32
asked Dec 30 '18 at 23:48
aNameLikeAnyOtheraNameLikeAnyOther
32
asked Dec 30 '18 at 23:48
aNameLikeAnyOtheraNameLikeAnyOther
32
32
$begingroup$
The second Venn should have some region unshaded. If that part is shrunk to nothing, result should look like first Venn.
$endgroup$
– coffeemath
Dec 31 '18 at 0:05
$begingroup$
But what about the rest of the universe in the second Venn? That still looks different from the first one
$endgroup$
– aNameLikeAnyOther
Dec 31 '18 at 0:12
$begingroup$
What is 'the rest of the universe'? You seem to be using two different ways of thinking about things inconsistently. First expressing the implication relationship by placing one circle inside the other, and then expressing the other relationship by shading. Try expressing the implication in terms of shading. (Also remember that $Pto Q$ holds when $P$ is false).
$endgroup$
– spaceisdarkgreen
Dec 31 '18 at 0:38
$begingroup$
By the rest of the universe I meant all the shaded area outside both the P and Q (in the second Venn, (~P or Q) one.) So there are two kinds of Venn diagrams: with and without shading? The one without shading is used express subset relationship, and the one with shading is for demonstrating boolean functions?
$endgroup$
– aNameLikeAnyOther
Dec 31 '18 at 2:17
add a comment |
$begingroup$
The second Venn should have some region unshaded. If that part is shrunk to nothing, result should look like first Venn.
$endgroup$
– coffeemath
Dec 31 '18 at 0:05
$begingroup$
But what about the rest of the universe in the second Venn? That still looks different from the first one
$endgroup$
– aNameLikeAnyOther
Dec 31 '18 at 0:12
$begingroup$
What is 'the rest of the universe'? You seem to be using two different ways of thinking about things inconsistently. First expressing the implication relationship by placing one circle inside the other, and then expressing the other relationship by shading. Try expressing the implication in terms of shading. (Also remember that $Pto Q$ holds when $P$ is false).
$endgroup$
– spaceisdarkgreen
Dec 31 '18 at 0:38
$begingroup$
By the rest of the universe I meant all the shaded area outside both the P and Q (in the second Venn, (~P or Q) one.) So there are two kinds of Venn diagrams: with and without shading? The one without shading is used express subset relationship, and the one with shading is for demonstrating boolean functions?
$endgroup$
– aNameLikeAnyOther
Dec 31 '18 at 2:17
$begingroup$
The second Venn should have some region unshaded. If that part is shrunk to nothing, result should look like first Venn.
$endgroup$
– coffeemath
Dec 31 '18 at 0:05
$begingroup$
The second Venn should have some region unshaded. If that part is shrunk to nothing, result should look like first Venn.
$endgroup$
– coffeemath
Dec 31 '18 at 0:05
$begingroup$
But what about the rest of the universe in the second Venn? That still looks different from the first one
$endgroup$
– aNameLikeAnyOther
Dec 31 '18 at 0:12
$begingroup$
But what about the rest of the universe in the second Venn? That still looks different from the first one
$endgroup$
– aNameLikeAnyOther
Dec 31 '18 at 0:12
$begingroup$
What is 'the rest of the universe'? You seem to be using two different ways of thinking about things inconsistently. First expressing the implication relationship by placing one circle inside the other, and then expressing the other relationship by shading. Try expressing the implication in terms of shading. (Also remember that $Pto Q$ holds when $P$ is false).
$endgroup$
– spaceisdarkgreen
Dec 31 '18 at 0:38
$begingroup$
What is 'the rest of the universe'? You seem to be using two different ways of thinking about things inconsistently. First expressing the implication relationship by placing one circle inside the other, and then expressing the other relationship by shading. Try expressing the implication in terms of shading. (Also remember that $Pto Q$ holds when $P$ is false).
$endgroup$
– spaceisdarkgreen
Dec 31 '18 at 0:38
$begingroup$
By the rest of the universe I meant all the shaded area outside both the P and Q (in the second Venn, (~P or Q) one.) So there are two kinds of Venn diagrams: with and without shading? The one without shading is used express subset relationship, and the one with shading is for demonstrating boolean functions?
$endgroup$
– aNameLikeAnyOther
Dec 31 '18 at 2:17
$begingroup$
By the rest of the universe I meant all the shaded area outside both the P and Q (in the second Venn, (~P or Q) one.) So there are two kinds of Venn diagrams: with and without shading? The one without shading is used express subset relationship, and the one with shading is for demonstrating boolean functions?
$endgroup$
– aNameLikeAnyOther
Dec 31 '18 at 2:17
add a comment |
1 Answer
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$begingroup$
To compare two Venn diagrams for a Boolean combination of P,Q one needs for each to shade (or check off) the regions of each diagram for which the Boolean is true. In case of $P implies Q,$ this means three regions: (P and Q), ((not P) and Q), and ((not P and (not Q)). That is, all regions checked except a region (if any exists in the diagram) for (P and (not Q)).
In the first diagram, with P inside Q, there isn't a region corresponding to (P and (not Q)), but the remaining three regions should be checked off, as noted above.
In the second (usual) Venn diagram, there are also exactly three regions to check off: All regions except that lying inside P and outside Q.
So in each diagram three regions are checked, and the same labels for each in the two diagrams. A confusing part in this may be that in the P inside Q case all regions are checked (there being only three such regions) but in the second "usual" Venn there are four regions with only three checked off. But the two diagrams correspond.
answered Dec 31 '18 at 6:19
coffeemathcoffeemath
2,8571415
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add a comment |
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$begingroup$
To compare two Venn diagrams for a Boolean combination of P,Q one needs for each to shade (or check off) the regions of each diagram for which the Boolean is true. In case of $P implies Q,$ this means three regions: (P and Q), ((not P) and Q), and ((not P and (not Q)). That is, all regions checked except a region (if any exists in the diagram) for (P and (not Q)).
In the first diagram, with P inside Q, there isn't a region corresponding to (P and (not Q)), but the remaining three regions should be checked off, as noted above.
In the second (usual) Venn diagram, there are also exactly three regions to check off: All regions except that lying inside P and outside Q.
So in each diagram three regions are checked, and the same labels for each in the two diagrams. A confusing part in this may be that in the P inside Q case all regions are checked (there being only three such regions) but in the second "usual" Venn there are four regions with only three checked off. But the two diagrams correspond.
answered Dec 31 '18 at 6:19
coffeemathcoffeemath
2,8571415
$endgroup$
add a comment |
$begingroup$
To compare two Venn diagrams for a Boolean combination of P,Q one needs for each to shade (or check off) the regions of each diagram for which the Boolean is true. In case of $P implies Q,$ this means three regions: (P and Q), ((not P) and Q), and ((not P and (not Q)). That is, all regions checked except a region (if any exists in the diagram) for (P and (not Q)).
In the first diagram, with P inside Q, there isn't a region corresponding to (P and (not Q)), but the remaining three regions should be checked off, as noted above.
In the second (usual) Venn diagram, there are also exactly three regions to check off: All regions except that lying inside P and outside Q.
So in each diagram three regions are checked, and the same labels for each in the two diagrams. A confusing part in this may be that in the P inside Q case all regions are checked (there being only three such regions) but in the second "usual" Venn there are four regions with only three checked off. But the two diagrams correspond.
answered Dec 31 '18 at 6:19
coffeemathcoffeemath
2,8571415
$endgroup$
add a comment |
$begingroup$
To compare two Venn diagrams for a Boolean combination of P,Q one needs for each to shade (or check off) the regions of each diagram for which the Boolean is true. In case of $P implies Q,$ this means three regions: (P and Q), ((not P) and Q), and ((not P and (not Q)). That is, all regions checked except a region (if any exists in the diagram) for (P and (not Q)).
In the first diagram, with P inside Q, there isn't a region corresponding to (P and (not Q)), but the remaining three regions should be checked off, as noted above.
In the second (usual) Venn diagram, there are also exactly three regions to check off: All regions except that lying inside P and outside Q.
So in each diagram three regions are checked, and the same labels for each in the two diagrams. A confusing part in this may be that in the P inside Q case all regions are checked (there being only three such regions) but in the second "usual" Venn there are four regions with only three checked off. But the two diagrams correspond.
answered Dec 31 '18 at 6:19
coffeemathcoffeemath
2,8571415
$endgroup$
To compare two Venn diagrams for a Boolean combination of P,Q one needs for each to shade (or check off) the regions of each diagram for which the Boolean is true. In case of $P implies Q,$ this means three regions: (P and Q), ((not P) and Q), and ((not P and (not Q)). That is, all regions checked except a region (if any exists in the diagram) for (P and (not Q)).
In the first diagram, with P inside Q, there isn't a region corresponding to (P and (not Q)), but the remaining three regions should be checked off, as noted above.
In the second (usual) Venn diagram, there are also exactly three regions to check off: All regions except that lying inside P and outside Q.
So in each diagram three regions are checked, and the same labels for each in the two diagrams. A confusing part in this may be that in the P inside Q case all regions are checked (there being only three such regions) but in the second "usual" Venn there are four regions with only three checked off. But the two diagrams correspond.
answered Dec 31 '18 at 6:19
coffeemathcoffeemath
2,8571415
answered Dec 31 '18 at 6:19
coffeemathcoffeemath
2,8571415
answered Dec 31 '18 at 6:19
coffeemathcoffeemath
2,8571415
answered Dec 31 '18 at 6:19
coffeemathcoffeemath
2,8571415
2,8571415
add a comment |
add a comment |
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$begingroup$
The second Venn should have some region unshaded. If that part is shrunk to nothing, result should look like first Venn.
$endgroup$
– coffeemath
Dec 31 '18 at 0:05
$begingroup$
But what about the rest of the universe in the second Venn? That still looks different from the first one
$endgroup$
– aNameLikeAnyOther
Dec 31 '18 at 0:12
$begingroup$
What is 'the rest of the universe'? You seem to be using two different ways of thinking about things inconsistently. First expressing the implication relationship by placing one circle inside the other, and then expressing the other relationship by shading. Try expressing the implication in terms of shading. (Also remember that $Pto Q$ holds when $P$ is false).
$endgroup$
– spaceisdarkgreen
Dec 31 '18 at 0:38
$begingroup$
By the rest of the universe I meant all the shaded area outside both the P and Q (in the second Venn, (~P or Q) one.) So there are two kinds of Venn diagrams: with and without shading? The one without shading is used express subset relationship, and the one with shading is for demonstrating boolean functions?
$endgroup$
– aNameLikeAnyOther
Dec 31 '18 at 2:17