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MyWikiBiz, Author Your Legacy — Tuesday April 08, 2025
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'''Exercises.''' Express the following in functional terms:+
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−<blockquote>
−<math>(\forall x \in X)(p(x) \Rightarrow q(x))</math>
−</blockquote>
−
−<blockquote>
−<math>\prod_{x \in X} (p_x (q_x)) = 1</math>
−</blockquote>
−
−This is just the form <math>\operatorname{All}\ p\ \operatorname{are}\ q,</math> already covered here:
−
−: [[Directory:Jon_Awbrey/Papers/Functional_Logic_:_Quantification_Theory#Application_of_Higher_Order_Propositions_to_Quantification_Theory|Application of Higher Order Propositions to Quantification Theory]]
−
−Need to think a little more about the proposition <math>p \Rightarrow q</math> as a boolean function of type <math>\mathbb{B}^2 \to \mathbb{B}</math> and the corresponding higher order proposition of type <math>(\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B}.</math>
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===Functional quantifiers===
===Functional quantifiers===
−====Tables====
−====Exercise 1====
−{| align="center" border="1" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
{| align="center" border="1" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
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|}<br>
|}<br>
−====Exercise 2====
+====Exercises====
+Express the following formulas in functional terms.
+=====Exercise 1=====
+<blockquote>
+<math>(\forall x \in X)(p(x) \Rightarrow q(x))</math>
+</blockquote>
+<blockquote>
+<math>\prod_{x \in X} (p_x (q_x)) = 1</math>
+</blockquote>
+This is just the form <math>\operatorname{All}\ p\ \operatorname{are}\ q,</math> already covered here:
+: [[Directory:Jon_Awbrey/Papers/Functional_Logic_:_Quantification_Theory#Application_of_Higher_Order_Propositions_to_Quantification_Theory|Application of Higher Order Propositions to Quantification Theory]]
+Need to think a little more about the proposition <math>p \Rightarrow q</math> as a boolean function of type <math>\mathbb{B}^2 \to \mathbb{B}</math> and the corresponding higher order proposition of type <math>(\mathbb{B}^2 \to \mathbb{B}) \to \mathbb{B}.</math>
+=====Exercise 2=====
<blockquote>
<blockquote>
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</blockquote>
</blockquote>
−====Exercise 3====
+=====Exercise 3=====
<blockquote>
<blockquote>
<math>(\forall x \in X)(Px \Rightarrow Qx) \lor (\forall x \in X)(Qx \Rightarrow Px)</math>
<math>(\forall x \in X)(Px \Rightarrow Qx) \lor (\forall x \in X)(Qx \Rightarrow Px)</math>
</blockquote>
</blockquote>
