9

How can I represent "If A then B" in a diagram? I thought it would be a simple subset like $A ⊂ B$. However this material says that

If $A$ then $B$ $=$ $A^c ∪ B$.

Now I am confused.

Hyunjun Kim
  • 93
  • 2
    Very good question. You're not wrong, and neither is your material, but they are just representing different things; see my answer. – user21820 Jul 14 '15 at 03:27
  • Didn't you ask for a diagram? You modified the question, but still asked for a diagram! – Moti Jul 14 '15 at 06:41
  • @Moti Yes, that is because I misunderstood "If $A$ then $B$" is the same as $A⊂B$. My question might be incorrect because of my ignorance. Sorry ;) – Hyunjun Kim Jul 14 '15 at 06:47

4 Answers4

6

You want to construct the set $\;\{x\mid x\in A \to x\in B\}\;$.

Then by implication equivalence this is $\;\{x\mid x\not\in A \vee x\in B\}\;$.

Which is simply $\;A^\complement \cup B\;$.

This is the set of all elements that, if they're in A then they're in B

The statement $A\subseteq B$ is not a set. It is a relation. It is the statement that $y\in A \implies y\in B$.

In the specific case that $A$ is a subset of $B$, then there is no element that is not in $A^\complement \cup B$.

So if you wanted to represent the statement "if $A$ then $B$", you could have $A$ as a subset of $B$.

But if you wanted to represent all elements that "if in $A$ then in $B$" you would use the union: $A^\complement\cup B$.

Graham Kemp
  • 129,094
5

Both representations you mentioned represent different things. "If $A$ then $B$" is a statement that is either true or false if each of $A,B$ are either true or false. The statement is true under certain conditions, namely as long as we do not have both $A$ being true and $B$ being false, which is equivalent to having either $A$ being false or $B$ being true, which is $(\neg A) \lor B$. Now if you represent $A,B$ by the sets of conditions where $A,B$ respectively are true, then $(\neg A) \lor B$ would be represented by the set $A^c \cup B$. This is why the venn diagram for implication is often given to have $A^c \cup B$ shaded.

On the other hand, your interpretation is not wrong either, but just about a different thing. The assertion "If $A$ then $B$" says that every condition under which $A$ is true is also a condition under which $B$ is true. Hence the set of conditions where $A$ is true is a subset of the set of conditions where $B$ is true.

So your diagram corresponds to the assertion itself (what is the case when it is true), while the other diagram corresponds to the truth value of the assertion (the shaded region is when it is true).

user21820
  • 57,693
  • 9
  • 98
  • 256
  • Where is the diagram? – Moti Jul 14 '15 at 06:42
  • @Moti: The asker already knows both diagrams but was asking why his diagram was different from that in the material he was referring to. I don't think it is necessary to provide the diagrams again. In any case, I didn't downvote your answer if you are wondering, because it's not wrong. However, it did not answer the underlying confusion so that's probably why other people downvoted it. – user21820 Jul 14 '15 at 06:48
  • If this is the case, than the q should be edited, since he asks for the diagram - I just read again the question. But what is important that the asker will be satisfied, and if he is than fine with me. And about some not so smart guy here... that's life. In any case thanks for the clarification. – Moti Jul 15 '15 at 07:34
  • @Moti: It is "then" not "than". Anyway I do not need to explicitly give the diagram; all I need to do is to resolve the confusion the asker or other people who have the same question have. – user21820 Jul 15 '15 at 07:45
1

"If $A$, then $B$" can be represented logically by the complement of its negation.

Specifically, "NOT (If $A$, then $B$)" $\leftrightarrow$ "$A$ and not $B$", so that "If $A$, then $B$" is the same as "NOT ($A$ and not $B$)" which, using De Morgan's laws, is "(not A) OR B". That is, you want $A^c\cup B$, as they said.

Terra Hyde
  • 1,102
0

It is quite simple - the external shape is B and A is fully included

enter image description here

layman
  • 20,191
Moti
  • 1,928
  • 3
    This doesn't account for $B$ that are not $A$. – Terra Hyde Jul 14 '15 at 03:01
  • 1
    Your textual description is still wrong! – user21820 Jul 14 '15 at 03:10
  • This graph is correct, if one notes that the depicted graph is an Euler diagram. A Venn diagram would look different, since the latter also includes as unshaded areas in which "if A then B" is false, namely areas of A which are outside of B. – Max Mar 11 '22 at 18:54
  • I found this post while searching google about this picture specifically. Turn out both are correct, except the picture that you drew here is usually considered to be an Euler diagram and not venn-driagram . But both carry the same information. If it is in A, then it is in B. If it is not in B, then it is not in A. If it is not in A, it may or may not be in B. All these statements apply to both diagrams. – Chhangsreng P Jan 16 '23 at 19:47