query: Why $p \leftrightarrow q$ is equivalent to $(p \wedge q) \vee (\neg p \wedge \neg q)$? Without using the truth table
positive_passages.docid: 48213
positive_passages.text: proving logical equivalence $(P \leftrightarrow Q) \equiv (P \wedge Q) \vee (\neg P \wedge \neg Q)$
negative_passages.docid: 5293
negative_passages.text: Show that the logic $ \ \forall x \exists y \ P(x,y) \ $ is not logically equivalent to $ \ \exists x \forall y \ P(x,y) \ $
negative_passages.docid: 58028
negative_passages.text: Semisimplicity is equivalent to each simple left module is projective?
negative_passages.docid: 70117
negative_passages.text: Show that $(p \to q) \lor (q \to p)$ is a tautology
negative_passages.docid: 9282
negative_passages.text: Prove using natural deduction $((P \land Q) \rightarrow R) \vdash (P \rightarrow R) \lor (Q\rightarrow R)$
negative_passages.docid: 79407
negative_passages.text: How to solve ~(P → Q) : P & ~Q by natural deduction?
negative_passages.docid: 75381
negative_passages.text: How can we reason about "if P then Q" or "P only if Q" statements in propositional logic?
negative_passages.docid: 67709
negative_passages.text: Stuck with proof for $\forall A\forall B(\mathcal{P}(A)\cup\mathcal{P}(B)=\mathcal{P}(A\cup B)\rightarrow A\subseteq B \vee B\subseteq A)$
negative_passages.docid: 88104
negative_passages.text: Prove "$(p \lor q) \land (\neg p \lor r)) \rightarrow (q \lor r)$" is a tautology
negative_passages.docid: 35236
negative_passages.text: Proof $p \vdash q \Rightarrow p$
negative_passages.docid: 72034
negative_passages.text: Truth table and the meaning of $\oplus$ in propositional logic
negative_passages.docid: 115004
negative_passages.text: If $x\notin\mathbb Q$, then $\left|x-\frac{p}{q}\right|<\frac{1}{q^2}$ for infinitely many $\frac{p}{q}$?
negative_passages.docid: 105815
negative_passages.text: Does proving (second countable) $\Rightarrow$ (Lindelöf) require the axiom of choice?
negative_passages.docid: 93342
negative_passages.text: Show that $\neg$ and $\wedge$ form a functionally complete collection of logical operators.
negative_passages.docid: 93410
negative_passages.text: How to prove ~ (~P & ~Q) : P ∨ Q by natural deduction
negative_passages.docid: 93040
negative_passages.text: $f$ is surjective iff it has a right inverse: using the axiom of choice and errors in ProofWiki
negative_passages.docid: 26678
negative_passages.text: The meaning of $\rightsquigarrow$ in math?