1
vote
1answer
61 views

Thinking logically instead of Venn diagrams

I hit upon the following identity while reading the book How to Prove: $$(A \cup B) \backslash B \subseteq A$$ Now if I solve this logically I can reduce this like this: $$ \begin{gather*} x \in (A ...
0
votes
1answer
135 views

Can you conclude that A = B if A, B, and C are sets such that…

a. A ∪ C = B ∪ C b. A ∩ C = B ∩ C c. A ∩ C = B ∩ C and A ∪ C = B ∪ C My method of solving this was to convert everything to propositional logic, then to solve it to show that none of the above are ...
0
votes
4answers
65 views

Symmetric difference equality

Something I was thinking about earlier: If $A\triangle B=A \triangle C$, does $B=C$? Where $\triangle$ is symmetric difference. My intuition is telling me no, but I can't seem to think of an example ...
2
votes
2answers
46 views

Set logic to propositional logic

How would you convert set logic to propositional logic? In particular, I'm not sure how to handle converting $\subseteq$ For example: $$A-(\bar{B} \cup \bar{C}) \subseteq B \cap C$$ My attempt at ...
0
votes
1answer
313 views

Help to understand the proof of the cardinality of a Cartesian Product

I was reading the proof of this theorem: The proof is as follows: I understand the entire proof except for the conclusion: Because $h$ is bijective $\dots$ Why is it sufficient to show ...
0
votes
4answers
167 views

How to prove two sets are mutually exclusive without using a Venn diagram?

If $E_1$ and $E_2$ are subsets of $S$, then $E_1$ and $E_2 \cap \overline{E_1}$ are mutually exclusive, where $\overline{E_1}$ is the complement of $E_1$. Is there a way to prove this without drawing ...
4
votes
2answers
147 views

Cardinality of tautologies for propositional logic

I'm wondering how many tautologies there are in propositional logic. I'm thinking that it must be at least countable, since ($P_{1} \wedge P_{2} \wedge \cdots P_{n}) \models P_{i}$ should be a ...
0
votes
1answer
55 views

Cardinality of Distinct Hilbert Systems with Detachment

Let us consider all formulas T of classical propositional logic which are tautologies up to simple substitution of variables where a variable can get simply substituted for another variable if and ...
3
votes
2answers
98 views

Representing $A \rightarrow B$ as $A \supseteq B$ [duplicate]

I know that many people like to think of elementary logic in terms of Venn diagrams, i.e., elementary set theory. I have never found this representation useful, because I can never remember whether ...
6
votes
5answers
228 views

Overlap of Propositional Logic and Elementary Set Question

I am a bit stuck on a basic sets problem: We know from resolutions that $(p \lor q) \land (\neg p \lor r) \to q \lor r$. Use this fact to show that $(P \cup Q) \cap (\overline{P} \cup R) \subseteq ...
2
votes
1answer
1k views

Relation between XOR and Symmetric difference

I noticed that XOR and symmetric difference use the same symbol, $\oplus$. They also seem to have a similar structure: XOR: $(\neg P\wedge Q)\vee(P\wedge \neg Q)$ Symmetric Difference: $(A\cap ...