# what is the truth table for A subset B?

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.

• 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. – Shubham Johri Oct 22 '20 at 20:18

## 1 Answer

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 $$A \subseteq B$$ ?

• 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? – Shubham Johri Oct 23 '20 at 9:54
• @ShubhamJohri - NO. If $A$ is finite, it is enough to check all elements of $A$. – Mauro ALLEGRANZA Oct 23 '20 at 10:08
• 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 $p(x)\wedge q(x)$...? – Shubham Johri Oct 23 '20 at 10:10
• @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. – Mauro ALLEGRANZA Oct 23 '20 at 10:12