-
+
+
+
- Mathematics without infinitely small, continuous mathematical objects. The mathematics of finite sets.
We should avoid variables when forming statements, as they may change the logical value.
-
-
- \(2=7\) statement -
- \(x=5\) not a statement +
- $2=7$ statement +
- $x=5$ not a statement
-In logic we do not use the equals sign, we use the equivalence sign \(\equiv\).
+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.).
@@ -180,147 +182,147 @@
The operations done on propositional variables are called propositional connectives.
-
-
- 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 \((1011)\)) +
- 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 $(1011)$)
-
-
- \(\equiv \neg p \lor q\) +
- $\equiv \neg p \lor q$
Not necessarily connectives but unary operations:
-
-
- Negation: Denoted by ~, \(\neg\) or NOT, negates the one input \((10)\). +
- Negation: Denoted by ~, $\neg$ or NOT, negates the one input $(10)$.
-A (propositional) formula is a "properly constructed" logical expression.
+A (propositional) formula is a “properly constructed” logical expression.
-
-
- 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: \(p \land (q \lor r) \equiv (p \land q) \lor (p \land r)\) +
- 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.
-
-
- \(\neg(B \lor C)\) can be written as \(\neg B \land \neg C\) +
- $\neg(B \lor C)$ can be written as $\neg B \land \neg C$
-Double negation law: \(\neg(\neg p) \equiv p\)
+Double negation law: $\neg(\neg p) \equiv p$
-De Morgan's laws: \(\neg(p \land q) \equiv \neg p \lor \neg q\) and \(\neg(p \lor q) \equiv \neg p \land \neg q\).
+De Morgan’s laws: $\neg(p \land q) \equiv \neg p \lor \neg q$ and $\neg(p \lor q) \equiv \neg p \land \neg q$.
-If and only if (iff): \(p \iff p \equiv (p \implies q) \land (q \implies p)\)
+If and only if (iff): $p \iff p \equiv (p \implies q) \land (q \implies p)$
Contraposition law:
-
-
- \((p \implies q) \equiv (\neg q \implies \neg p)\) prove by contraposition +
- $(p \implies q) \equiv (\neg q \implies \neg p)$ prove by contraposition
-
-
- \((p \implies q) \equiv (\neg p \lor q)\) -
- \((\neg q \implies \neg p) \equiv (\neg (\neg q) \lor (\neg p) \equiv (q \lor \neg p) \equiv (\neg p \lor q)\) +
- $(p \implies q) \equiv (\neg p \lor q)$ +
- $(\neg q \implies \neg p) \equiv (\neg (\neg q) \lor (\neg p) \equiv (q \lor \neg p) \equiv (\neg p \lor q)$
Contradiction law:
-
-
- \(p \lor \neg p \equiv 1\) and \(p \land \neg p \equiv 0\) +
- $p \lor \neg p \equiv 1$ and $p \land \neg p \equiv 0$
-Tautology: \(\phi (p, q, ... r)\) is a tautology iff \(\phi \equiv 1\)
+Tautology: $\phi (p, q, … r)$ is a tautology iff $\phi \equiv 1$
Sets
-
-
We will consider subsets of universal set \(\mathbb X\)
+We will consider subsets of universal set $\mathbb X$
-
-
- \(2^\mathbb X = \{ A : A \subseteq \mathbb X\}\) -
- \(2^\mathbb X = P(\mathbb X)\) -
- All 2 object subsets of \(\mathbb X\): \(P_2(\mathbb X)\) +
- $2^\mathbb X = { A : A \subseteq \mathbb X}$ +
- $2^\mathbb X = P(\mathbb X)$ +
- All 2 object subsets of $\mathbb X$: $P_2(\mathbb X)$
-\(A \subset B \equiv\) every element of A is an element of B \(\equiv \{x \in \mathbb X : x \in A \implies x \in B\}\)
+$A \subset B \equiv$ every element of A is an element of B $\equiv {x \in \mathbb X : x \in A \implies x \in B}$
Operations on sets:
-
-
- 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 - \(A'\) - \(A' = \{ x \in \mathbb X : \neg (x \in A) \}\) +
- 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 - $A’$ - $A’ = { x \in \mathbb X : \neg (x \in A) }$
-
-
- If \(x = \{ 1 \}\) then \(x' = \emptyset\) +
- If $x = { 1 }$ then $x’ = \emptyset$
-Equality of sets: \(A = B\) iff \(x \in \mathbb X : (x \in A \iff x \in B)\)
+Equality of sets: $A = B$ iff $x \in \mathbb X : (x \in A \iff x \in B)$
Difference of sets:
-
-
- \(A \setminus B = \{ x \in \mathbb X : x \in A \land x \notin B \} = A \cap B'\) -
- Symmetric difference: \(A \div B = (A \setminus B) \cup (B \setminus A)\) +
- $A \setminus B = { x \in \mathbb X : x \in A \land x \notin B } = A \cap B’$ +
- Symmetric difference: $A \div B = (A \setminus B) \cup (B \setminus A)$
Laws of set algebra:
-
-
- \(A \cup B = B \cup A , A \cap B = B \cap A\) -
- \((A \cup B) \cup C = A \cup (B \cup C), (A \cap B) \cap C = A \cap (B \cap C)\) -
- \((A \cap (B \cup C) = (A \cap B) \cup (A \cap C)\) 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 -
- \(A \cup A' = \mathbb X, A \cap A' = \emptyset\) +
- $A \cup B = B \cup A , A \cap B = B \cap A$ +
- $(A \cup B) \cup C = A \cup (B \cup C), (A \cap B) \cap C = A \cap (B \cap C)$ +
- $(A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$ 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 (\(\{ \}\))
+Note: ${ \emptyset } \neq \emptyset$, one is a set with one element, one is the empty set, no elements (${ }$)
-Quip: \(\{ x \in \mathbb R : x^2 = -1\} = \emptyset\)
+Quip: ${ x \in \mathbb R : x^2 = -1} = \emptyset$
Quantifiers
-
-
- \(\phi\) - prepositional function: yields only true or false value -
- \(\forall\) means "for all" and \(\exists\) means "there exists" +
- $\phi$ - prepositional function: yields only true or false value +
- $\forall$ means “for all” and $\exists$ means “there exists” -
\(\forall\):
+$\forall$:
-
-
- Shorthand for \(\land\) e.g. \((\forall x \in \{ 1, 2, ... 10 \}) x > 0 \equiv 1 > 0 \land 2 > 0 \land ... 10 > 0\) +
- Shorthand for $\land$ e.g. $(\forall x \in { 1, 2, … 10 }) x > 0 \equiv 1 > 0 \land 2 > 0 \land … 10 > 0$
-\(\exists\):
+$\exists$:
-
-
- Shorthand for \(\lor\) e.g. \((\exists x \in \{ 1, 2, ... 10 \}) x > 5 \equiv 1 > 5 \lor 2 > 5 \lor ... 10 > 5\) +
- 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
+$\neg \forall \equiv \exists$, vice versa
With quantifiers we can write logical statements e.g.
-
-
- \((\forall x \in \mathbb{R}) (\forall y \in \mathbb{R}) x > y\) is a statement and is false -
- \((\forall x) (\exists y) x > y\) is true -
- shortcut: \((\exists x, y) \equiv (\exists x) (\exists y)\) +
- $(\forall x \in \mathbb{R}) (\forall y \in \mathbb{R}) x > y$ 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:
-
-
- \((\forall x \in \mathbb{X}) (\phi(x)) \equiv \{ p \in \mathbb{X} : \phi(p) \} = \mathbb{X}\) -
- \((\exists x \in \mathbb{X}) (\phi(x)) \equiv \{ q \in \mathbb{X} : \phi(q) \} \neq \emptyset\) -
- \((\exists x \in \mathbb{X}) (\neg \phi(x)) \equiv \neg ( \{ p \in \mathbb{X} : \phi(p) \} = \mathbb{X} )\) +
- $(\forall x \in \mathbb{X}) (\phi(x)) \equiv { p \in \mathbb{X} : \phi(p) } = \mathbb{X}$ +
- $(\exists x \in \mathbb{X}) (\phi(x)) \equiv { q \in \mathbb{X} : \phi(q) } \neq \emptyset$ +
- $(\exists x \in \mathbb{X}) (\neg \phi(x)) \equiv \neg ( { p \in \mathbb{X} : \phi(p) } = \mathbb{X} )$
Order of quantifiers matters.
@@ -332,53 +334,53 @@
Cartesian product:
-
-
- \(A \times B = \{ (p, q) : p \in A \land q \in B \}\) +
- $A \times B = { (p, q) : p \in A \land q \in B }$
-Def: A relation \(R\) on a set \(\mathbb X\) is a subset of \(\mathbb X \times \mathbb X\) (\(R \subseteq \mathbb X \times \mathbb X\))
+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 \(f()\): \(\{ (x, f(x) : x \in Dom(f) \}\)
+Graph of a function $f()$: ${ (x, f(x) : x \in Dom(f) }$
Properties of:
-
-
- 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: \((\forall x, y \in \mathbb X) (x R y \land y R x \implies x = y)\) +
- 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:
-
-
- 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 \([x]_R = \{ y \in \mathbb X : x R y \}\) +
- 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 $[x]_R = { y \in \mathbb X : x R y }$
-
-
- Every \(x \in \mathbb X\) belongs to the equivalence class of some element \(a\). -
- \((\forall x, y \in \mathbb X) ([x] \cap [y] \neq \emptyset \iff [x] = [y])\) +
- Every $x \in \mathbb X$ belongs to the equivalence class of some element $a$. +
- $(\forall x, y \in \mathbb X) ([x] \cap [y] \neq \emptyset \iff [x] = [y])$
Partitions
-
-
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:
+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:
-
-
- \((\forall x \in \mathbb X) (\exists j \in \mathbb I) (x \in A_j)\) -
- \((\forall i, j \in \mathbb I) (i \neq j \implies A_i \cap A_j = \emptyset)\) +
- $(\forall x \in \mathbb X) (\exists j \in \mathbb I) (x \in A_j)$ +
- $(\forall i, j \in \mathbb I) (i \neq j \implies A_i \cap A_j = \emptyset)$
-\(\{ A_i \}_{i \in \mathbb I}\) is a partition iff there exists an equivalence relation \(R\) on \(\mathbb X\) such that:
+${ Ai }{i \in \mathbb I}$ is a partition iff there exists an equivalence relation $R$ on $\mathbb X$ such that:
-
-
- \((\forall i \in \mathbb I) (\exists x \in \mathbb X) A_i = [x]_R\) -
- \((\forall x \in \mathbb X) (\exists j \in \mathbb I) [x] = A_j\) +
- $(\forall i \in \mathbb I) (\exists x \in \mathbb X) A_i = [x]_R$ +
- $(\forall x \in \mathbb X) (\exists j \in \mathbb I) [x] = A_j$
-The quotient set: \(\mathbb X / R = \{ [a] : a \in \mathbb X \}\)
+The quotient set: $\mathbb X / R = { [a] : a \in \mathbb X }$
Partial orders
-
-
- \(\mathbb X\) is a set, \(R \subseteq \mathbb X \times \mathbb X\) +
- $\mathbb X$ is a set, $R \subseteq \mathbb X \times \mathbb X$ -
Def: \(R\) is a partial order on \(\mathbb X\) iff \(R\) is:
+Def: $R$ is a partial order on $\mathbb X$ iff $R$ is:
- Reflexive @@ -398,24 +400,24 @@
- Transitive
-Def: \(m \in \mathbb X\) is said to be:
+Def: $m \in \mathbb X$ is said to be:
-
-
- maximal element in \((\mathbb X, \preccurlyeq)\) iff \((\forall a \in \mathbb X) m \preccurlyeq a \implies m = a\) -
- largest iff \((\forall a \in \mathbb X) (a \preccurlyeq m)\) -
- minimal iff \((\forall a \in \mathbb X) (a \preccurlyeq m \implies a = m)\) -
- smallest iff \((\forall a \in \mathbb X) (m \preccurlyeq a)\) +
- maximal element in $(\mathbb X, \preccurlyeq)$ iff $(\forall a \in \mathbb X) m \preccurlyeq a \implies m = a$ +
- largest iff $(\forall a \in \mathbb X) (a \preccurlyeq m)$ +
- minimal iff $(\forall a \in \mathbb X) (a \preccurlyeq m \implies a = m)$ +
- smallest iff $(\forall a \in \mathbb X) (m \preccurlyeq a)$
-Def: A partial order \(R\) on \(\mathbb X\) is said to be "total" iff \((\forall x, y \in \mathbb X) (x R y \lor y R x)\)
+Def: A partial order $R$ on $\mathbb X$ is said to be “total” iff $(\forall x, y \in \mathbb X) (x R y \lor y R x)$
-Def: A subset \(B\) of \(\mathbb X\) is called a chain "chain" iff \(B\) is totally ordered by \(R\)
+Def: A subset $B$ of $\mathbb X$ is called a chain “chain” iff $B$ is totally ordered by $R$
-
-
- \(C(\mathbb X)\) - the set of all chains in \((\mathbb X, R)\) -
- A chain \(D\) in \((\mathbb X, R)\) is called a maximal chain iff \(D\) is a maximal element in \((C(\mathbb X), R)\) -
- \(K \subseteq \mathbb X\) is called an antichain in \((\mathbb X, R)\) iff \((\forall p, q \in K) (p R q \lor q R p \implies p = q)\) -
- Def: \(R\) is a partial order on \(\mathbb X\), \(R\) is called a well order iff \(R\) is a total order on \(X\) and every nonempty subset \(A\) of \(\mathbb X\) has the smallest element +
- $C(\mathbb X)$ - the set of all chains in $(\mathbb X, R)$ +
- A chain $D$ in $(\mathbb X, R)$ is called a maximal chain iff $D$ is a maximal element in $(C(\mathbb X), R)$ +
- $K \subseteq \mathbb X$ is called an antichain in $(\mathbb X, R)$ iff $(\forall p, q \in K) (p R q \lor q R p \implies p = q)$ +
- Def: $R$ is a partial order on $\mathbb X$, $R$ is called a well order iff $R$ is a total order on $X$ and every nonempty subset $A$ of $\mathbb X$ has the smallest element
Induction
-
-
- If \(\phi\) is a propositional function defined on \(\mathbb N\), if: +
- If $\phi$ is a propositional function defined on $\mathbb N$, if:
-
-
- \(\phi(1)\) -
- \((\forall n \geq 1) \phi(n) \implies \phi(n+1)\) -
- \((\forall k \geq 1) \phi(k)\) +
- $\phi(1)$ +
- $(\forall n \geq 1) \phi(n) \implies \phi(n+1)$ +
- $(\forall k \geq 1) \phi(k)$
Functions
-
-
- \(f: \mathbb X \to \mathbb Y\) +
- $f: \mathbb X \to \mathbb Y$ -
Def: \(f \subseteq \mathbb X \times \mathbb Y\) is said to be a function if:
+Def: $f \subseteq \mathbb X \times \mathbb Y$ is said to be a function if:
-
-
- \((\forall x \in \mathbb X)(\exists y \in \mathbb Y) (x, y) \in f(y = f(x))\) -
- \((\forall a \in \mathbb X)(\forall p, q \in \mathbb Y)((a, p) \in f \land (a, q) \in f \implies p = q)\) +
- $(\forall x \in \mathbb X)(\exists y \in \mathbb Y) (x, y) \in f(y = f(x))$ +
- $(\forall a \in \mathbb X)(\forall p, q \in \mathbb Y)((a, p) \in f \land (a, q) \in f \implies p = q)$
-Types of functions \(f: \mathbb X \to \mathbb Y\):
+Types of functions $f: \mathbb X \to \mathbb Y$:
-
-
- \(f\) is said to be an injection ( 1 to 1 function) iff \((\forall x_1, x_2 \in \mathbb X) x_1 \neq x_2 \implies f(x_1) \neq f(x_2)\) -
- \(f\) is said to be a surjection (onto function) iff \((\forall y \in \mathbb Y)(\exists x \in \mathbb X) f(x) = y\) -
- If \(f^{-1}\) is a function from \(\mathbb Y \to \mathbb X\) then \(f^{-1}\) is called the inverse function for \(f\) +
- $f$ is said to be an injection ( 1 to 1 function) iff $(\forall x_1, x_2 \in \mathbb X) x_1 \neq x_2 \implies f(x_1) \neq f(x_2)$ +
- $f$ is said to be a surjection (onto function) iff $(\forall y \in \mathbb Y)(\exists x \in \mathbb X) f(x) = y$ +
- If $f^{-1}$ is a function from $\mathbb Y \to \mathbb X$ then $f^{-1}$ is called the inverse function for $f$
-
-
- Fact: \(f^{-1}\) is a function iff \(f\) is a bijection (1 to 1 and onto) +
- Fact: $f^{-1}$ is a function iff $f$ is a bijection (1 to 1 and onto)
-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: \(f^{-1}(\mathbb A)\)
+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: $f^{-1}(\mathbb A)$