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
1,1811318
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
1,1811318
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
1,1811318
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
1,1811318
asked Dec 30 '18 at 23:48
aNameLikeAnyOtheraNameLikeAnyOther
32
edited Dec 31 '18 at 1:12
Gaby Alfonso
1,1811318
asked Dec 30 '18 at 23:48
aNameLikeAnyOtheraNameLikeAnyOther
32
edited Dec 31 '18 at 1:12
Gaby Alfonso
1,1811318
edited Dec 31 '18 at 1:12
Gaby Alfonso
1,1811318
edited Dec 31 '18 at 1:12
Gaby Alfonso
1,1811318
1,1811318
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
1
active
oldest
votes
$begingroup$
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
2,9071415
$endgroup$
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%2f3057291%2ftwo-different-venn-diagrams-for-p-rightarrow-q-implication-subset-diagram-an%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
$begingroup$
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
2,9071415
$endgroup$
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
2,9071415
$endgroup$
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
2,9071415
$endgroup$
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
2,9071415
answered Dec 31 '18 at 6:19
coffeemathcoffeemath
2,9071415
answered Dec 31 '18 at 6:19
coffeemathcoffeemath
2,9071415
answered Dec 31 '18 at 6:19
coffeemathcoffeemath
2,9071415
2,9071415
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%2f3057291%2ftwo-different-venn-diagrams-for-p-rightarrow-q-implication-subset-diagram-an%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
$begingroup$
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