By p⊕q, also written p xor q, I mean either $p$ or $q$ but not both; and I use the symbol $\sim $ to negate propositions.
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3Either $p$ or $q$ but not both is called exclusive or (xor). To check whether what you are asking is true try making the truth tables for both the expressions and compare them. – Giorgos Giapitzakis Aug 12 '21 at 23:42
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2This is known as Exclusive Or which is sometimes denoted $p \oplus q$ – WaveX Aug 12 '21 at 23:43
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1@GiorgosGiapitzakis Just did, mate. Thank you! – Aug 12 '21 at 23:44
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@WaveX didn't know, thank you! – Aug 12 '21 at 23:44
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Did you mean propositions where you typed prepositions in the title? – J. W. Tanner Aug 12 '21 at 23:54
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@J.W.Tanner Hehehe, yeah. – Aug 13 '21 at 00:02
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What have you tried? – Doug Spoonwood Aug 13 '21 at 03:05
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@DougSpoonwood Out of Giorgos's Advice I made the truth tables for both expressions and realised they're logically equivalent. :) – Aug 13 '21 at 03:09
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Since $$((p\to\lnot q)\land(\lnot p\to q))\:\leftrightarrow\:((p\lor q)\land\lnot(p\land q))$$ is a tautology, the assumptions $$(p\implies\lnot q) \text{ and } (\lnot p\implies q)$$ jointly indeed suffice to entail the conclusion $$p ⊕ q.$$
ryang
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