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Let me explain we know that: XOR (exclusive-or) ≡ symmetric difference = (A-B) ∪ (B-A).

we know that: XNOR (¬XOR i.e. negation of XOR) ≡ ↔ (Bi-condontional statement) = A=B iff (if and only if) B=A.

we know that: A-B (set difference) ≡ PΛ¬Q ≡ ¬(P→Q).

we know that: A∩B ≡ PΛQ

we know that: A∪B ≡ PνQ

I know this because I worked it out last night here are the truth tables in this PDF: https://www.dropbox.com/s/nc8201ccwwis4hi/truth%20tables.pdf?dl=0

now comes the fun part I have absolutely no idea what the hell are the truth tables for these:

  • A ⊆ B
  • A ⊂ B
  • A ⊇ B
  • A ⊃ B

also what are the truth tables for negation of these sets:

  • ¬(A ⊆ B)
  • ¬(A ⊂ B)
  • ¬(A ⊇ B)
  • ¬(A ⊃ B)

Please Somebody help my brain is wrecked and I can't think any more about this stuff. Let me know if there is anything you need I'd be happy to try and answer.

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    $\begingroup$ You make truth tables for boolean expressions involving truth values $0,1$. $A\subset B$ does not make sense when $A,B$ are truth values. $\endgroup$ – Shubham Johri Oct 22 '20 at 20:18
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You are misreading the "equivalences" above.

It is not true that $A \cap B \equiv P \land Q$.

$P \land Q$ is a statement whose truth value is either True or False.

$A \cap B$ is the "name" of a set: the intersection of sets $A$ and $B$.

But there is a link between the two: $A \cap B$ is defined by formula $(x \in A \land x \in B)$, that means:

$\forall x \ [x \in (A \cap B) \leftrightarrow (x \in A \land x \in B)]$.

Things are similar for $\subseteq$ (and all the "derived" ones), but there is a difference: $A \subseteq B$ is not a "name" for a set but a statement involving two sets, and its definition is:

$A \subseteq B \leftrightarrow \forall x ( x \in A \to x \in B)$.

In conclusion: there is a link between boolean connectives: $\lnot, \land \lor, \to$ and set operations: complement, intersection, union, inclusion, but we cannot simply equate them.


Having said that, in principle when can use truth tables to "check $A \subseteq B$ when $A$ and $B$ are finite sets.

But what is the benefit for e.g. $A = \{ 0, 1, 3, 7 \}$ and $B = \mathbb N$ to write the truth table with four lines and two columns to verify te statement $A \subseteq B$ ?

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  • $\begingroup$ A truth table for the statement $A\subseteq B$ gives us the value of the statement $0/1$ for each value of $A$ and $B$. Since $A,B$ are not $0/1$, we have infinite possibilities even when both sets are finite. How can you draw the truth table? $\endgroup$ – Shubham Johri Oct 23 '20 at 9:54
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    $\begingroup$ @ShubhamJohri - NO. If $A$ is finite, it is enough to check all elements of $A$. $\endgroup$ – Mauro ALLEGRANZA Oct 23 '20 at 10:08
  • $\begingroup$ You have selected particular sets for $A,B$. That is like selecting $p(x)=1,q(x)=1$ and concluding $p(x)\wedge q(x)=1$ is the truth table for the statement $p(x)\wedge q(x)$...? $\endgroup$ – Shubham Johri Oct 23 '20 at 10:10
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    $\begingroup$ @ShubhamJohri - IF $A$ is finite, whatever it is, it is enough to check the elements in $A$. For the example above, $\pi \notin A$; thus $\pi \in A \to \pi \in \mathbb N$ is TRUE. $\endgroup$ – Mauro ALLEGRANZA Oct 23 '20 at 10:12

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