So this time I'm trying find the equivalent forms of some simple propositions, but I can only make the equivalent formulas using $\to$ and $\bot$ as the only connectives. One is $p \lor q$ and the other's $p \leftrightarrow q$. I've been trying to figure this out for hours but nothing's coming up.
Discrete math: Logical equivalence using limited connectives
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discrete-mathematics
logic
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0I am somewhat confused by what you are trying to do. Are you just trying to find equivalent forms of propositions or are you trying to do so using only certain symbols? – 2017-02-12
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0Trying to find equivalent forms using just the two symbols specified. – 2017-02-12
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0$\lnot p$ must be defined as $p \to \bot$. – 2017-02-12
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0$p \lor q$ must be defined as $\lnot p \to q$ i.e. as $(p \to \bot) \to q$. – 2017-02-12
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0$p↔q$ must be defined as $(p→q)∧(q→p)$. – 2017-02-12
1 Answers
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$\lnot p$ must be defined as:
$p \to \bot$.
$p∨q$ must be defined as $¬p→q$, i.e. as $(p→⊥)→q$.
$p∧q$ must be defined as $¬(p→¬q)$, i.e. as $(p→(q \to \bot)) \to \bot$.
$p↔q$ must be defined as $(p→q)∧(q→p)$, i.e. as:
$((p→q)→((q→p) \to \bot)) \to \bot$.
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0Thank you. I kept trying the find a small propositional statement to solve these because I was using a truth table. – 2017-02-12