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\(2=7\) statement \(x=5\) not a statement In logic we do not use the equals sign, we use the equivalence sign \(\equiv\).
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< li > Mathematics without infinitely small, continuous mathematical objects. The mathematics of finite sets.< / li >
< / ul >
< h2 id = "propositional-calculus" > Propositional calculus< / h2 >
< ul >
< li > Comes from the linguistic concept that things can be either true or false.< / li >
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< li > < p > We should avoid variables when forming statements, as they may change the logical value.< / p >
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< ul >
< li > < span class = "math" > \(2=7\)< / span > statement< / li >
< li > < span class = "math" > \(x=5\)< / span > not a statement< / li >
< / ul > < / li >
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< li > < p > In logic we do not use the equals sign, we use the equivalence sign < span class = "math" > \(\equiv\)< / span > .< / p > < / li >
< li > < p > Logical values (booleans) are denoted by either 0 or 1 (or t, f, etc.).< / p > < / li >
< li > < p > When doing logic, we use propositional variables (e.g. p, q, r).< / p >
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< ul >
< li > Can be either < strong > true< / strong > or < strong > false< / strong > .< / li >
< / ul > < / li >
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< li > < p > The operations done on propositional variables are called propositional connectives.< / p >
< ul >
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< li > Conjunction: < span class = "math" > \(p \land q\)< / span > is only true if both p and q are true < span class = "math" > \((0001)\)< / span > < / li >
< li > Disjunction: < span class = "math" > \(p \lor q\)< / span > is only false if both p and q are false < span class = "math" > \((0111)\)< / span > < / li >
< li > Implication (material conditional): < span class = "math" > \(p \implies q\)< / span > is false only if p is true and q is false (truth table < span class = "math" > \((1011)\)< / span > )
< ul >
< li > < span class = "math" > \(\equiv \neg p \lor q\)< / span > < / li >
< / ul > < / li >
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< / ul > < / li >
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< li > < p > Not necessarily connectives but unary operations:< / p >
< ul >
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< li > Negation: Denoted by ~, < span class = "math" > \(\neg\)< / span > or NOT, negates the one input < span class = "math" > \((10)\)< / span > .< / li >
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< / ul > < / li >
< li > < p > A (propositional) formula is a " properly constructed" logical expression.< / p >
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< ul >
< li > e.g. < span class = "math" > \(\neg[(p \lor q)] \land r\)< / span > < / li >
< li > < span class = "math" > \((p \land)\)< / span > is not a formula, as < span class = "math" > \(\land\)< / span > requires 2 variables.< / li >
< li > Logical equivalence: < span class = "math" > \(\phi(p, q, k) \equiv \psi(p, q, k)\)< / span > , logical value of < span class = "math" > \(\phi\)< / span > is equal to logical value of < span class = "math" > \(\psi\)< / span > .< / li >
< li > Commutativity: < span class = "math" > \(p \land q \equiv q \land p\)< / span > < / li >
< li > Associativity: < span class = "math" > \((p \land q) \land r \equiv p \land (q \land r)\)< / span > < / li >
< li > Distributivity: < span class = "math" > \(p \land (q \lor r) \equiv (p \land q) \lor (p \land r)\)< / span > < / li >
< li > Conjunctive normal form: every formula can be written as a conjunction of one or more disjunctions.
< ul >
< li > < span class = "math" > \(\neg(B \lor C)\)< / span > can be written as < span class = "math" > \(\neg B \land \neg C\)< / span > < / li >
< / ul > < / li >
< / ul > < / li >
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< li > < p > Double negation law: < span class = "math" > \(\neg(\neg p) \equiv p\)< / span > < / p > < / li >
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< li > < p > De Morgan's laws: < span class = "math" > \(\neg(p \land q) \equiv \neg p \lor \neg q\)< / span > and < span class = "math" > \(\neg(p \lor q) \equiv \neg p \land \neg q\)< / span > .< / p > < / li >
< li > < p > If and only if (< em > iff< / em > ): < span class = "math" > \(p \iff p \equiv (p \implies q) \land (q \implies p)\)< / span > < / p > < / li >
< li > < p > Contraposition law:< / p >
< ul >
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< li > < span class = "math" > \((p \implies q) \equiv (\neg q \implies \neg p)\)< / span > prove by contraposition
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< ul >
< li > < span class = "math" > \((p \implies q) \equiv (\neg p \lor q)\)< / span > < / li >
< li > < span class = "math" > \((\neg q \implies \neg p) \equiv (\neg (\neg q) \lor (\neg p) \equiv (q \lor \neg p) \equiv (\neg p \lor q)\)< / span > < / li >
< / ul > < / li >
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< / ul > < / li >
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< li > < p > Contradiction law:< / p >
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< ul >
< li > < span class = "math" > \(p \lor \neg p \equiv 1\)< / span > and < span class = "math" > \(p \land \neg p \equiv 0\)< / span > < / li >
< / ul > < / li >
< li > < p > Tautology: < span class = "math" > \(\phi (p, q, ... r)\)< / span > is a tautology < em > iff< / em > < span class = "math" > \(\phi \equiv 1\)< / span > < / p > < / li >
< / ul >
< h2 id = "sets" > Sets< / h2 >
< ul >
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< li > < p > We will consider subsets of universal set < span class = "math" > \(\mathbb X\)< / span > < / p >
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< ul >
< li > < span class = "math" > \(2^\mathbb X = \{ A : A \subseteq \mathbb X\}\)< / span > < / li >
< li > < span class = "math" > \(2^\mathbb X = P(\mathbb X)\)< / span > < / li >
< li > All 2 object subsets of < span class = "math" > \(\mathbb X\)< / span > : < span class = "math" > \(P_2(\mathbb X)\)< / span > < / li >
< / ul > < / li >
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< li > < p > < span class = "math" > \(A \subset B \equiv\)< / span > every element of A is an element of B < span class = "math" > \(\equiv \{x \in \mathbb X : x \in A \implies x \in B\}\)< / span > < / p > < / li >
< li > < p > Operations on sets:< / p >
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< ul >
< li > Union - < span class = "math" > \(\cup\)< / span > - < span class = "math" > \(A \cup B = \{ x \in \mathbb X : x \in A \lor x \in B \}\)< / span > < / li >
< li > Intersection - < span class = "math" > \(\cap\)< / span > - < span class = "math" > \(A \cap B = \{ x \in \mathbb X : x \in A \land x \in B \}\)< / span > < / li >
< li > Complement - < span class = "math" > \(A'\)< / span > - < span class = "math" > \(A' = \{ x \in \mathbb X : \neg (x \in A) \}\)< / span >
< ul >
< li > If < span class = "math" > \(x = \{ 1 \}\)< / span > then < span class = "math" > \(x' = \emptyset\)< / span > < / li >
< / ul > < / li >
< / ul > < / li >
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< li > < p > Equality of sets: < span class = "math" > \(A = B\)< / span > iff < span class = "math" > \(x \in \mathbb X : (x \in A \iff x \in B)\)< / span > < / p > < / li >
< li > < p > Difference of sets:< / p >
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< ul >
< li > < span class = "math" > \(A \setminus B = \{ x \in \mathbb X : x \in A \land x \notin B \} = A \cap B'\)< / span > < / li >
< li > Symmetric difference: < span class = "math" > \(A \div B = (A \setminus B) \cup (B \setminus A)\)< / span > < / li >
< / ul > < / li >
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< li > < p > Laws of set algebra:< / p >
< ul >
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< li > < span class = "math" > \(A \cup B = B \cup A , A \cap B = B \cap A\)< / span > < / li >
< li > < span class = "math" > \((A \cup B) \cup C = A \cup (B \cup C), (A \cap B) \cap C = A \cap (B \cap C)\)< / span > < / li >
< li > < span class = "math" > \((A \cap (B \cup C) = (A \cap B) \cup (A \cap C)\)< / span > vice versa< / li >
< li > < span class = "math" > \(A \cap \emptyset, A \cap \mathbb X = A, A \cup \emptyset = A, A \cup \mathbb X = \mathbb X\)< / span > < / li >
< li > < span class = "math" > \((A \cup B)' = A' \cap B'\)< / span > vice versa< / li >
< li > < span class = "math" > \(A \cup A' = \mathbb X, A \cap A' = \emptyset\)< / span > < / li >
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< / ul > < / li >
< li > < p > Note: < span class = "math" > \(\{ \emptyset \} \neq \emptyset\)< / span > , one is a set with one element, one is the empty set, no elements (< span class = "math" > \(\{ \}\)< / span > )< / p > < / li >
< li > < p > Quip: < span class = "math" > \(\{ x \in \mathbb R : x^2 = -1\} = \emptyset\)< / span > < / p > < / li >
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< / ul >
< h2 id = "quantifiers" > Quantifiers< / h2 >
< ul >
< li > < span class = "math" > \(\phi\)< / span > - prepositional function: yields only true or false value< / li >
< li > < span class = "math" > \(\forall\)< / span > means " for all" and < span class = "math" > \(\exists\)< / span > means " there exists" < / li >
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< li > < p > < span class = "math" > \(\forall\)< / span > :< / p >
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< ul >
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< li > Shorthand for < span class = "math" > \(\land\)< / span > e.g. < span class = "math" > \((\forall x \in \{ 1, 2, ... 10 \}) x > 0 \equiv 1 > 0 \land 2 > 0 \land ... 10 > 0\)< / span > < / li >
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< / ul > < / li >
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< li > < p > < span class = "math" > \(\exists\)< / span > :< / p >
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< ul >
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< li > Shorthand for < span class = "math" > \(\lor\)< / span > e.g. < span class = "math" > \((\exists x \in \{ 1, 2, ... 10 \}) x > 5 \equiv 1 > 5 \lor 2 > 5 \lor ... 10 > 5\)< / span > < / li >
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< / ul > < / li >
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< li > < p > < span class = "math" > \(\neg \forall \equiv \exists\)< / span > , vice versa< / p > < / li >
< li > < p > With quantifiers we can write logical statements e.g.< / p >
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< ul >
< li > < span class = "math" > \((\forall x \in \mathbb{R}) (\forall y \in \mathbb{R}) x > y\)< / span > is a statement and is false< / li >
< li > < span class = "math" > \((\forall x) (\exists y) x > y\)< / span > is true< / li >
< li > shortcut: < span class = "math" > \((\exists x, y) \equiv (\exists x) (\exists y)\)< / span > < / li >
< / ul > < / li >
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< li > < p > Quantifiers can be expressed in set language, sort of a definition in terms of sets:< / p >
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< ul >
< li > < span class = "math" > \((\forall x \in \mathbb{X}) (\phi(x)) \equiv \{ p \in \mathbb{X} : \phi(p) \} = \mathbb{X}\)< / span > < / li >
< li > < span class = "math" > \((\exists x \in \mathbb{X}) (\phi(x)) \equiv \{ q \in \mathbb{X} : \phi(q) \} \neq \emptyset\)< / span > < / li >
< li > < span class = "math" > \((\exists x \in \mathbb{X}) (\neg \phi(x)) \equiv \neg ( \{ p \in \mathbb{X} : \phi(p) \} = \mathbb{X} )\)< / span > < / li >
< / ul > < / li >
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< li > < p > Order of quantifiers matters.< / p > < / li >
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< / ul >
< h2 id = "relations" > Relations< / h2 >
< ul >
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< li > < p > Cartesian product:< / p >
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< ul >
< li > < span class = "math" > \(A \times B = \{ (p, q) : p \in A \land q \in B \}\)< / span > < / li >
< / ul > < / li >
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< li > < p > Def: A relation < span class = "math" > \(R\)< / span > on a set < span class = "math" > \(\mathbb X\)< / span > is a subset of < span class = "math" > \(\mathbb X \times \mathbb X\)< / span > (< span class = "math" > \(R \subseteq \mathbb X \times \mathbb X\)< / span > )< / p > < / li >
< li > < p > Graph of a function < span class = "math" > \(f()\)< / span > : < span class = "math" > \(\{ (x, f(x) : x \in Dom(f) \}\)< / span > < / p > < / li >
< li > < p > Properties of:< / p >
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< ul >
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< li > Reflexivity: < span class = "math" > \((\forall x \in \mathbb X ) (x, x) \in R \equiv (\forall x \in \mathbb X) x R x\)< / span > < / li >
< li > Symmetricity: < span class = "math" > \([ (\forall x, y \in \mathbb X) (x, y) \in R \implies (y, x) \in R) ] \equiv [ (\forall x, y \in \mathbb X) ( x R y \implies y R x) ]\)< / span > < / li >
< li > Transitivity: < span class = "math" > \((\forall x, y, z \in \mathbb X) (x R y \land y R z \implies x R z)\)< / span > < / li >
< li > Antisymmetricity: < span class = "math" > \((\forall x, y \in \mathbb X) (x R y \land y R x \implies x = y)\)< / span > < / li >
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< / ul > < / li >
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< li > < p > Equivalence relations:< / p >
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< ul >
< li > Def: < span class = "math" > \(R \subseteq \mathbb X \times \mathbb X\)< / span > is said to be an equivalence relation < em > iff< / em > < span class = "math" > \(R\)< / span > is reflexive, symmetric and transitive.< / li >
< li > Congruence modulo n: < span class = "math" > \(p R q \equiv n | p - q\)< / span > < / li >
< li > Def R - and equivalence relation of < span class = "math" > \(\mathbb X\)< / span > : The < em > equivalence class< / em > of an element < span class = "math" > \(x \in \mathbb X\)< / span > is the set < span class = "math" > \([x]_R = \{ y \in \mathbb X : x R y \}\)< / span >
< ul >
< li > Every < span class = "math" > \(x \in \mathbb X\)< / span > belongs to the equivalence class of some element < span class = "math" > \(a\)< / span > .< / li >
< li > < span class = "math" > \((\forall x, y \in \mathbb X) ([x] \cap [y] \neq \emptyset \iff [x] = [y])\)< / span > < / li >
< / ul > < / li >
< / ul > < / li >
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< li > < p > Partitions< / p >
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< ul >
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< li > < p > A partition is a set containing subsets of some set < span class = "math" > \(\mathbb X\)< / span > such that their collective symmetric difference equals < span class = "math" > \(\mathbb X\)< / span > . A partition of is a set < span class = "math" > \(\{ A_i: i \in \mathbb I \land A_i \subseteq \mathbb X \}\)< / span > such that:< / p >
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< ul >
< li > < span class = "math" > \((\forall x \in \mathbb X) (\exists j \in \mathbb I) (x \in A_j)\)< / span > < / li >
< li > < span class = "math" > \((\forall i, j \in \mathbb I) (i \neq j \implies A_i \cap A_j = \emptyset)\)< / span > < / li >
< / ul > < / li >
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< li > < p > < span class = "math" > \(\{ A_i \}_{i \in \mathbb I}\)< / span > is a partition < em > iff< / em > there exists an equivalence relation < span class = "math" > \(R\)< / span > on < span class = "math" > \(\mathbb X\)< / span > such that:< / p >
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< ul >
< li > < span class = "math" > \((\forall i \in \mathbb I) (\exists x \in \mathbb X) A_i = [x]_R\)< / span > < / li >
< li > < span class = "math" > \((\forall x \in \mathbb X) (\exists j \in \mathbb I) [x] = A_j\)< / span > < / li >
< / ul > < / li >
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< li > < p > The quotient set: < span class = "math" > \(\mathbb X / R = \{ [a] : a \in \mathbb X \}\)< / span > < / p > < / li >
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