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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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$2=7$ 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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$2=7$ 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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< ul >
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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 >
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< li > $2=7$ statement< / li >
< li > $x=5$ not a statement< / li >
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< / ul > < / li >
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< li > < p > In logic we do not use the equals sign, we use the equivalence sign $\equiv$.< / p > < / li >
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< 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: $p \land q$ is only true if both p and q are true $(0001)$< / li >
< li > Disjunction: $p \lor q$ is only false if both p and q are false $(0111)$< / li >
< li > Implication (material conditional): $p \implies q$ is false only if p is true and q is false (truth table $(1011)$)
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< ul >
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< li > $\equiv \neg p \lor q$< / li >
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< / 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 ~, $\neg$ or NOT, negates the one input $(10)$.< / li >
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< / ul > < / li >
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< li > < p > A (propositional) formula is a “ properly constructed” logical expression.< / p >
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< ul >
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< li > e.g. $\neg[(p \lor q)] \land r$< / li >
< li > $(p \land)$ is not a formula, as $\land$ requires 2 variables.< / li >
< li > Logical equivalence: $\phi(p, q, k) \equiv \psi(p, q, k)$, logical value of $\phi$ is equal to logical value of $\psi$.< / li >
< li > Commutativity: $p \land q \equiv q \land p$< / li >
< li > Associativity: $(p \land q) \land r \equiv p \land (q \land r)$< / li >
< li > Distributivity: $p \land (q \lor r) \equiv (p \land q) \lor (p \land r)$< / li >
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< li > Conjunctive normal form: every formula can be written as a conjunction of one or more disjunctions.
< ul >
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< li > $\neg(B \lor C)$ can be written as $\neg B \land \neg C$< / li >
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< / ul > < / li >
< / ul > < / li >
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< li > < p > Double negation law: $\neg(\neg p) \equiv p$< / p > < / li >
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< li > < p > De Morgan’ s laws: $\neg(p \land q) \equiv \neg p \lor \neg q$ and $\neg(p \lor q) \equiv \neg p \land \neg q$.< / p > < / li >
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< li > < p > If and only if (< em > iff< / em > ): $p \iff p \equiv (p \implies q) \land (q \implies p)$< / p > < / li >
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< li > < p > Contraposition law:< / p >
< ul >
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< li > $(p \implies q) \equiv (\neg q \implies \neg p)$ prove by contraposition
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< ul >
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< li > $(p \implies q) \equiv (\neg p \lor q)$< / li >
< li > $(\neg q \implies \neg p) \equiv (\neg (\neg q) \lor (\neg p) \equiv (q \lor \neg p) \equiv (\neg p \lor q)$< / li >
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< / ul > < / li >
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< / ul > < / li >
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< li > < p > Contradiction law:< / p >
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< ul >
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< li > $p \lor \neg p \equiv 1$ and $p \land \neg p \equiv 0$< / li >
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< / ul > < / li >
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< li > < p > Tautology: $\phi (p, q, … r)$ is a tautology < em > iff< / em > $\phi \equiv 1$< / p > < / li >
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< / ul >
< h2 id = "sets" > Sets< / h2 >
< ul >
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< li > < p > We will consider subsets of universal set $\mathbb X$< / p >
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< ul >
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< li > $2^\mathbb X = { A : A \subseteq \mathbb X}$< / li >
< li > $2^\mathbb X = P(\mathbb X)$< / li >
< li > All 2 object subsets of $\mathbb X$: $P_2(\mathbb X)$< / li >
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< / ul > < / li >
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< li > < p > $A \subset B \equiv$ every element of A is an element of B $\equiv {x \in \mathbb X : x \in A \implies x \in B}$< / p > < / li >
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< li > < p > Operations on sets:< / p >
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< ul >
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< li > Union - $\cup$ - $A \cup B = { x \in \mathbb X : x \in A \lor x \in B }$< / li >
< li > Intersection - $\cap$ - $A \cap B = { x \in \mathbb X : x \in A \land x \in B }$< / li >
< li > Complement - $A’ $ - $A’ = { x \in \mathbb X : \neg (x \in A) }$
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< ul >
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< li > If $x = { 1 }$ then $x’ = \emptyset$< / li >
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< / ul > < / li >
< / ul > < / li >
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< li > < p > Equality of sets: $A = B$ iff $x \in \mathbb X : (x \in A \iff x \in B)$< / p > < / li >
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< li > < p > Difference of sets:< / p >
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< ul >
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< li > $A \setminus B = { x \in \mathbb X : x \in A \land x \notin B } = A \cap B’ $< / li >
< li > Symmetric difference: $A \div B = (A \setminus B) \cup (B \setminus A)$< / li >
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< / ul > < / li >
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< li > < p > Laws of set algebra:< / p >
< ul >
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< li > $A \cup B = B \cup A , A \cap B = B \cap A$< / li >
< li > $(A \cup B) \cup C = A \cup (B \cup C), (A \cap B) \cap C = A \cap (B \cap C)$< / li >
< li > $(A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$ vice versa< / li >
< li > $A \cap \emptyset, A \cap \mathbb X = A, A \cup \emptyset = A, A \cup \mathbb X = \mathbb X$< / li >
< li > $(A \cup B)’ = A’ \cap B’ $ vice versa< / li >
< li > $A \cup A’ = \mathbb X, A \cap A’ = \emptyset$< / li >
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< / ul > < / li >
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< li > < p > Note: ${ \emptyset } \neq \emptyset$, one is a set with one element, one is the empty set, no elements (${ }$)< / p > < / li >
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< li > < p > Quip: ${ x \in \mathbb R : x^2 = -1} = \emptyset$< / p > < / li >
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< / ul >
< h2 id = "quantifiers" > Quantifiers< / h2 >
< ul >
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< li > $\phi$ - prepositional function: yields only true or false value< / li >
< li > $\forall$ means “ for all” and $\exists$ means “ there exists” < / li >
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< li > < p > $\forall$:< / p >
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< ul >
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< li > Shorthand for $\land$ e.g. $(\forall x \in { 1, 2, … 10 }) x > 0 \equiv 1 > 0 \land 2 > 0 \land … 10 > 0$< / li >
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< / ul > < / li >
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< li > < p > $\exists$:< / p >
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< ul >
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< li > Shorthand for $\lor$ e.g. $(\exists x \in { 1, 2, … 10 }) x > 5 \equiv 1 > 5 \lor 2 > 5 \lor … 10 > 5$< / li >
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< / ul > < / li >
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< li > < p > $\neg \forall \equiv \exists$, vice versa< / p > < / li >
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< li > < p > With quantifiers we can write logical statements e.g.< / p >
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< ul >
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< li > $(\forall x \in \mathbb{R}) (\forall y \in \mathbb{R}) x > y$ is a statement and is false< / li >
< li > $(\forall x) (\exists y) x > y$ is true< / li >
< li > shortcut: $(\exists x, y) \equiv (\exists x) (\exists y)$< / li >
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< / 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 >
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< li > $(\forall x \in \mathbb{X}) (\phi(x)) \equiv { p \in \mathbb{X} : \phi(p) } = \mathbb{X}$< / li >
< li > $(\exists x \in \mathbb{X}) (\phi(x)) \equiv { q \in \mathbb{X} : \phi(q) } \neq \emptyset$< / li >
< li > $(\exists x \in \mathbb{X}) (\neg \phi(x)) \equiv \neg ( { p \in \mathbb{X} : \phi(p) } = \mathbb{X} )$< / li >
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< / 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 >
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< li > $A \times B = { (p, q) : p \in A \land q \in B }$< / li >
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< / ul > < / li >
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< li > < p > Def: A relation $R$ on a set $\mathbb X$ is a subset of $\mathbb X \times \mathbb X$ ($R \subseteq \mathbb X \times \mathbb X$)< / p > < / li >
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< li > < p > Graph of a function $f()$: ${ (x, f(x) : x \in Dom(f) }$< / p > < / li >
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< li > < p > Properties of:< / p >
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< ul >
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< li > Reflexivity: $(\forall x \in \mathbb X ) (x, x) \in R \equiv (\forall x \in \mathbb X) x R x$< / li >
< li > Symmetricity: $[ (\forall x, y \in \mathbb X) (x, y) \in R \implies (y, x) \in R) ] \equiv [ (\forall x, y \in \mathbb X) ( x R y \implies y R x) ]$< / li >
< li > Transitivity: $(\forall x, y, z \in \mathbb X) (x R y \land y R z \implies x R z)$< / li >
< li > Antisymmetricity: $(\forall x, y \in \mathbb X) (x R y \land y R x \implies x = y)$< / li >
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< / ul > < / li >
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< li > < p > Equivalence relations:< / p >
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< ul >
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< li > Def: $R \subseteq \mathbb X \times \mathbb X$ is said to be an equivalence relation < em > iff< / em > $R$ is reflexive, symmetric and transitive.< / li >
< li > Congruence modulo n: $p R q \equiv n | p - q$< / li >
< li > Def R - and equivalence relation of $\mathbb X$: The < em > equivalence class< / em > of an element $x \in \mathbb X$ is the set $[x]_R = { y \in \mathbb X : x R y }$
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< ul >
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< li > Every $x \in \mathbb X$ belongs to the equivalence class of some element $a$.< / li >
< li > $(\forall x, y \in \mathbb X) ([x] \cap [y] \neq \emptyset \iff [x] = [y])$< / li >
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< / 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 $\mathbb X$ such that their collective symmetric difference equals $\mathbb X$. A partition of is a set ${ A_i: i \in \mathbb I \land A_i \subseteq \mathbb X }$ such that:< / p >
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< ul >
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< li > $(\forall x \in \mathbb X) (\exists j \in \mathbb I) (x \in A_j)$< / li >
< li > $(\forall i, j \in \mathbb I) (i \neq j \implies A_i \cap A_j = \emptyset)$< / li >
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< / ul > < / li >
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< li > < p > ${ A< em > i }< / em > {i \in \mathbb I}$ is a partition < em > iff< / em > there exists an equivalence relation $R$ on $\mathbb X$ such that:< / p >
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< ul >
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< li > $(\forall i \in \mathbb I) (\exists x \in \mathbb X) A_i = [x]_R$< / li >
< li > $(\forall x \in \mathbb X) (\exists j \in \mathbb I) [x] = A_j$< / li >
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< / ul > < / li >
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< li > < p > The quotient set: $\mathbb X / R = { [a] : a \in \mathbb X }$< / p > < / li >
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< / ul > < / li >
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< / ul >
< h2 id = "posets" > Posets< / h2 >
< ul >
< li > < p > Partial orders< / p >
< ul >
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< li > $\mathbb X$ is a set, $R \subseteq \mathbb X \times \mathbb X$< / li >
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< li > < p > Def: $R$ is a partial order on $\mathbb X$ iff $R$ is:< / p >
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< ul >
< li > Reflexive< / li >
< li > Antisymmetric< / li >
< li > Transitive< / li >
< / ul > < / li >
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< li > < p > Def: $m \in \mathbb X$ is said to be:< / p >
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< ul >
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< li > maximal element in $(\mathbb X, \preccurlyeq)$ iff $(\forall a \in \mathbb X) m \preccurlyeq a \implies m = a$< / li >
< li > largest iff $(\forall a \in \mathbb X) (a \preccurlyeq m)$< / li >
< li > minimal iff $(\forall a \in \mathbb X) (a \preccurlyeq m \implies a = m)$< / li >
< li > smallest iff $(\forall a \in \mathbb X) (m \preccurlyeq a)$< / li >
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< / ul > < / li >
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< li > < p > Def: A partial order $R$ on $\mathbb X$ is said to be < em > “ total” < / em > iff $(\forall x, y \in \mathbb X) (x R y \lor y R x)$< / p > < / li >
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< li > < p > Def: A subset $B$ of $\mathbb X$ is called a chain < em > “ chain” < / em > iff $B$ is totally ordered by $R$< / p >
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< ul >
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< li > $C(\mathbb X)$ - the set of all chains in $(\mathbb X, R)$< / li >
< li > A chain $D$ in $(\mathbb X, R)$ is called a maximal chain iff $D$ is a maximal element in $(C(\mathbb X), R)$< / li >
< li > $K \subseteq \mathbb X$ is called an antichain in $(\mathbb X, R)$ iff $(\forall p, q \in K) (p R q \lor q R p \implies p = q)$< / li >
< li > Def: $R$ is a partial order on $\mathbb X$, $R$ is called a < em > well< / em > order iff $R$ is a total order on $X$ and every nonempty subset $A$ of $\mathbb X$ has the smallest element< / li >
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< / ul > < / li >
< / ul > < / li >
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< h2 id = "induction" > Induction< / h2 >
< ul >
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< li > If $\phi$ is a propositional function defined on $\mathbb N$, if:
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< ul >
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< li > $\phi(1)$< / li >
< li > $(\forall n \geq 1) \phi(n) \implies \phi(n+1)$< / li >
< li > $(\forall k \geq 1) \phi(k)$< / li >
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< / ul > < / li >
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< / ul >
< h2 id = "functions" > Functions< / h2 >
< ul >
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< li > $f: \mathbb X \to \mathbb Y$< / li >
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< li > < p > Def: $f \subseteq \mathbb X \times \mathbb Y$ is said to be a function if:< / p >
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< ul >
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< li > $(\forall x \in \mathbb X)(\exists y \in \mathbb Y) (x, y) \in f(y = f(x))$< / li >
< li > $(\forall a \in \mathbb X)(\forall p, q \in \mathbb Y)((a, p) \in f \land (a, q) \in f \implies p = q)$< / li >
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< / ul > < / li >
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< li > < p > Types of functions $f: \mathbb X \to \mathbb Y$:< / p >
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< ul >
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< li > $f$ is said to be an injection ( 1 to 1 function) iff $(\forall x_1, x_2 \in \mathbb X) x_1 \neq x_2 \implies f(x_1) \neq f(x_2)$< / li >
< li > $f$ is said to be a surjection (onto function) iff $(\forall y \in \mathbb Y)(\exists x \in \mathbb X) f(x) = y$< / li >
< li > If $f^{-1}$ is a function from $\mathbb Y \to \mathbb X$ then $f^{-1}$ is called the inverse function for $f$
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< ul >
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< li > Fact: $f^{-1}$ is a function iff $f$ is a < em > bijection< / em > (1 to 1 and onto)< / li >
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< / ul > < / li >
< / ul > < / li >
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< li > < p > For some set $\mathbb A$ the image of $\mathbb A$ by $f$ is $f(\mathbb A) = { f(x) : x \in \mathbb A }$. We can also define the inverse of an image even when the function itself isn’ t invertible: $f^{-1}(\mathbb A)$< / p > < / li >
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