Let be $F=(A\land B)$ and $G=\neg(\neg A \lor \neg B)$. Which of the following statements are correct
$F=G, F\equiv G, \neg F=\neg G, \neg F\equiv\neg G$?
Is there a difference?
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$\begingroup$ Hello! We discourage posts that simply state a problem out of context, and expect the community to solve it. What are your thoughts? What have you tried? Where did you get stuck? Did you try writing out a truth table? Assuming you tried to solve it yourself and got stuck, it may be helpful if you wrote your thoughts and what you could not figure out. $\endgroup$– D.W. ♦Oct 26, 2015 at 0:32
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1 Answer
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Equality is a syntactic notion, equivalence is a semantic notion. Two expressions are equal if they are the same expression — in other words, an expression is only equal to itself. Two logical expressions are equivalent if they have the same truth value in every interpretation.
Two equal expressions are always equivalent, but the converse doesn't hold. For example, $A \equiv \lnot\lnot A$ but $A \neq \lnot\lnot A$.
answered Oct 25, 2015 at 11:13
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$\begingroup$ Thank you for firstly for your answer! So, the right statements would be $\neg F\equiv \neg G$ and $F\equiv G$. I've proved these with a truth table. $\endgroup$ Oct 25, 2015 at 11:37
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$\begingroup$ These identities are known as the de Morgan laws, and they also hold when switching AND and OR. $\endgroup$ Oct 25, 2015 at 12:22
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$\begingroup$ What could be said similarly about <-> and <=> ? I guess <=> is semantic, but is it synonymous of the triple bar equal sign? Is <-> syntactic or semantic? $\endgroup$– Hibou57Apr 30, 2020 at 15:46
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$\begingroup$ The symbols $\leftrightarrow$ and $\Leftrightarrow$ don't have any standard interpretation. You'll have to explain what these mean to you. $\endgroup$ Apr 30, 2020 at 15:48
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$\begingroup$ "Two expressions are equal if they are the same expression — in other words, an expression is only equal to itself." Perhaps you have something specific in mind when you say this, but its obviously not true outside of boolean formulas. Even within boolean formulas, equivalence satisfies all the 4 axioms of equality (reflexivity, symmetry, transitivity, and congruence) so ought to be treated as if it were equality over booleans $\endgroup$ Jan 4 at 2:47