Tidy up and regenerate
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@ -9,5 +9,5 @@ draft: false
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- name: Abdulkadir Furkan Şanlı
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- handle: abdulocracy
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- contact:
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- email: my handle at disroot dot org
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- irc (freenode): abdulocracy
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- email: my handle at disroot dot org
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- irc (freenode): abdulocracy
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@ -49,12 +49,13 @@ markup = "mmark"
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- $$(p \implies q) \equiv (\neg p \lor q)$$
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- $$(\neg q \implies \neg p) \equiv (\neg (\neg q) \lor (\neg p) \equiv (q \lor \neg p) \equiv (\neg p \lor q)$$
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- Contradiction law:
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- $$p \lor \neg p \equiv 1$$ and $$p \land \neg p \equiv 0$$
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- Contradiction law:
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- $$p \lor \neg p \equiv 1$$ and $$p \land \neg p \equiv 0$$
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- Tautology: $$\phi (p, q, ... r)$$ is a tautology *iff* $$\phi \equiv 1$$
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## Sets
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- We will consider subsets of universal set $$\mathbb X$$
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- $$2^\mathbb X = \{ A : A \subseteq \mathbb X\}$$
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- $$2^\mathbb X = P(\mathbb X)$$
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@ -84,12 +85,13 @@ markup = "mmark"
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- Quip: $$\{ x \in \mathbb R : x^2 = -1\} = \emptyset$$
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## Quantifiers
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- $$\phi$$ - prepositional function: yields only true or false value
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- $$\forall$$ means "for all" and $$\exists$$ means "there exists"
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- $$\forall$$
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- $$\forall$$:
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- Shorthand for $$\land$$ e.g. $$(\forall x \in \{ 1, 2, ... 10 \}) x > 0 \equiv 1 > 0 \land 2 > 0 \land ... 10 > 0$$
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- $$\exists$$
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- $$\exists$$:
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- Shorthand for $$\lor$$ e.g. $$(\exists x \in \{ 1, 2, ... 10 \}) x > 5 \equiv 1 > 5 \lor 2 > 5 \lor ... 10 > 5$$
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- $$\neg \forall \equiv \exists$$, vice versa
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@ -106,6 +108,7 @@ markup = "mmark"
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- Order of quantifiers matters.
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## Relations
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- Cartesian product:
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- $$A \times B = \{ (p, q) : p \in A \land q \in B \}$$
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