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