Mathematics without infinitely small, continuous mathematical objects. The mathematics of finite sets. Propositional calculus 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.
Mathematics without infinitely small, continuous mathematical objects. The mathematics of finite sets. Propositional calculus 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. (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.
$2=7$ 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.
<pubDate>Wed, 20 Nov 2019 00:00:00 +0000</pubDate>
<pubDate>Wed, 20 Nov 2019 00:00:00 +0000</pubDate>
<guid>https://abdulocra.cy/posts/eidma/</guid>
<guid>https://abdulocra.cy/posts/eidma/</guid>
<description>Mathematics without infinitely small, continuous mathematical objects. The mathematics of finite sets. Propositional calculus 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.
<description>Mathematics without infinitely small, continuous mathematical objects. The mathematics of finite sets. Propositional calculus 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. \(2=7\) 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.</description>
$2=7$ 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.</description>
<metaname="description"content="Mathematics without infinitely small, continuous mathematical objects. The mathematics of finite sets. Propositional calculus 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. \(2=7\) 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."/>
$2=7$ 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."/>
<metaname="twitter:description"content="Mathematics without infinitely small, continuous mathematical objects. The mathematics of finite sets. Propositional calculus 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. \(2=7\) 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."/>
$2=7$ 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." />
<metaproperty="og:description"content="Mathematics without infinitely small, continuous mathematical objects. The mathematics of finite sets. Propositional calculus 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. \(2=7\) 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."/>
$2=7$ 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." />
<li><spanclass="math inline">\(x=5\)</span> not a statement</li>
</ul></li>
</ul></li>
<li>In logic we do not use the equals sign, we use the equivalence sign <spanclass="math inline">\(\equiv\)</span>.</li>
<li><p>In logic we do not use the equals sign, we use the equivalence sign $\equiv$.</p></li>
<li>Logical values (booleans) are denoted by either 0 or 1 (or t, f, etc.).</li>
<li>When doing logic, we use propositional variables (e.g.p, q, r).
<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>
<ul>
<ul>
<li>Can be either <strong>true</strong> or <strong>false</strong>.</li>
<li>Can be either <strong>true</strong> or <strong>false</strong>.</li>
</ul></li>
</ul></li>
<li>The operations done on propositional variables are called propositional connectives.
<li><p>The operations done on propositional variables are called propositional connectives.</p>
<ul>
<ul>
<li>Conjunction: $p \land q$ is only true if both p and q are true $(0001)$</li>
<li>Conjunction: <spanclass="math inline">\(p \land q\)</span> is only true if both p and q are true <spanclass="math inline">\((0001)\)</span></li>
<li>Disjunction: $p \lor q$ is only false if both p and q are false $(0111)$</li>
<li>Disjunction: <spanclass="math inline">\(p \lor q\)</span> is only false if both p and q are false <spanclass="math inline">\((0111)\)</span></li>
<li>Implication (material conditional): $p \implies q$ is false only if p is true and q is false (truth table $(1011)$)
<li>Implication (material conditional): <spanclass="math inline">\(p \implies q\)</span> is false only if p is true and q is false (truth table <spanclass="math inline">\((1011)\)</span>)
<ul>
<ul>
<li>$\equiv \neg p \lor q$</li>
<li><spanclass="math inline">\(\equiv \neg p \lor q\)</span></li>
</ul></li>
</ul></li>
</ul></li>
</ul></li>
<li>Not necessarily connectives but unary operations:
<li><p>Not necessarily connectives but unary operations:</p>
<ul>
<ul>
<li>Negation: Denoted by ~, $\neg$ or NOT, negates the one input $(10)$.</li>
<li>Negation: Denoted by ~, <spanclass="math inline">\(\neg\)</span> or NOT, negates the one input <spanclass="math inline">\((10)\)</span>.</li>
</ul></li>
</ul></li>
<li>A (propositional) formula is a “properly constructed” logical expression.
<li><p>A (propositional) formula is a “properly constructed” logical expression.</p>
<li>$(p \land)$ is not a formula, as $\land$ requires 2 variables.</li>
<li><spanclass="math inline">\((p \land)\)</span> is not a formula, as <spanclass="math inline">\(\land\)</span> 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>Logical equivalence: <spanclass="math inline">\(\phi(p, q, k) \equiv \psi(p, q, k)\)</span>, logical value of <spanclass="math inline">\(\phi\)</span> is equal to logical value of <spanclass="math inline">\(\psi\)</span>.</li>
<li>$p \lor \neg p \equiv 1$ and $p \land \neg p \equiv 0$</li>
<li><spanclass="math inline">\(p \lor \neg p \equiv 1\)</span> and <spanclass="math inline">\(p \land \neg p \equiv 0\)</span></li>
</ul></li>
</ul></li>
<li><p>Tautology: <spanclass="math inline">\(\phi (p, q, ... r)\)</span> is a tautology <em>iff</em><spanclass="math inline">\(\phi \equiv 1\)</span></p></li>
<li><p>Tautology: $\phi (p, q, … r)$ is a tautology <em>iff</em> $\phi \equiv 1$</p></li>
</ul>
</ul>
<h2id="sets">Sets</h2>
<h2id="sets">Sets</h2>
<ul>
<ul>
<li><p>We will consider subsets of universal set $\mathbb X$</p>
<li>We will consider subsets of universal set <spanclass="math inline">\(\mathbb X\)</span>
<ul>
<ul>
<li>$2^\mathbb X = { A : A \subseteq \mathbb X}$</li>
<li><spanclass="math inline">\(2^\mathbb X = \{ A : A \subseteq \mathbb X\}\)</span></li>
<li>$2^\mathbb X = P(\mathbb X)$</li>
<li><spanclass="math inline">\(2^\mathbb X = P(\mathbb X)\)</span></li>
<li>All 2 object subsets of $\mathbb X$: $P_2(\mathbb X)$</li>
<li><spanclass="math inline">\(A \subset B \equiv\)</span> every element of A is an element of B <spanclass="math inline">\(\equiv \{x \in \mathbb X : x \in A \implies x \in B\}\)</span></li>
<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>
<li>Operations on sets:
<li><p>Operations on sets:</p>
<ul>
<ul>
<li>Union - $\cup$ - $A \cup B = { x \in \mathbb X : x \in A \lor x \in B }$</li>
<li>Union - <spanclass="math inline">\(\cup\)</span> - <spanclass="math inline">\(A \cup B = \{ x \in \mathbb X : x \in A \lor x \in B \}\)</span></li>
<li>Intersection - $\cap$ - $A \cap B = { x \in \mathbb X : x \in A \land x \in B }$</li>
<li>Intersection - <spanclass="math inline">\(\cap\)</span> - <spanclass="math inline">\(A \cap B = \{ x \in \mathbb X : x \in A \land x \in B \}\)</span></li>
<li>Complement - $A’$ - $A’ = { x \in \mathbb X : \neg (x \in A) }$
<li>Complement - <spanclass="math inline">\(A'\)</span> - <spanclass="math inline">\(A' = \{ x \in \mathbb X : \neg (x \in A) \}\)</span>
<ul>
<ul>
<li>If $x = { 1 }$ then $x’ = \emptyset$</li>
<li>Note: <spanclass="math inline">\(\{ \emptyset \} \neq \emptyset\)</span>, one is a set with one element, one is the empty set, no elements (<spanclass="math inline">\(\{ \}\)</span>)</li>
<li><p>Note: ${ \emptyset } \neq \emptyset$, one is a set with one element, one is the empty set, no elements (${ }$)</p></li>
<li>Quip: <spanclass="math inline">\(\{ x \in \mathbb R : x^2 = -1\} = \emptyset\)</span></li>
<li><p>Quip: ${ x \in \mathbb R : x^2 = -1} = \emptyset$</p></li>
</ul>
</ul>
<h2id="quantifiers">Quantifiers</h2>
<h2id="quantifiers">Quantifiers</h2>
<ul>
<ul>
<li>$\phi$ - prepositional function: yields only true or false value</li>
<li><spanclass="math inline">\(\phi\)</span> - prepositional function: yields only true or false value</li>
<li>$\forall$ means “for all” and $\exists$ means “there exists”</li>
<li><spanclass="math inline">\(\forall\)</span> means “for all” and <spanclass="math inline">\(\exists\)</span> means “there exists”</li>
<li><spanclass="math inline">\(\forall\)</span>:
<li><p>$\forall$:</p>
<ul>
<ul>
<li>Shorthand for $\land$ e.g. $(\forall x \in { 1, 2, … 10 }) x > 0 \equiv 1 > 0 \land 2 > 0 \land … 10 > 0$</li>
<li>Shorthand for <spanclass="math inline">\(\land\)</span> e.g.<spanclass="math inline">\((\forall x \in \{ 1, 2, ... 10 \}) x > 0 \equiv 1 > 0 \land 2 > 0 \land ... 10 > 0\)</span></li>
</ul></li>
</ul></li>
<li><spanclass="math inline">\(\exists\)</span>:
<li><p>$\exists$:</p>
<ul>
<ul>
<li>Shorthand for $\lor$ e.g. $(\exists x \in { 1, 2, … 10 }) x > 5 \equiv 1 > 5 \lor 2 > 5 \lor … 10 > 5$</li>
<li>Shorthand for <spanclass="math inline">\(\lor\)</span> e.g.<spanclass="math inline">\((\exists x \in \{ 1, 2, ... 10 \}) x > 5 \equiv 1 > 5 \lor 2 > 5 \lor ... 10 > 5\)</span></li>
<li>$(\exists x \in \mathbb{X}) (\neg \phi(x)) \equiv \neg ( { p \in \mathbb{X} : \phi(p) } = \mathbb{X} )$</li>
<li><spanclass="math inline">\((\exists x \in \mathbb{X}) (\neg \phi(x)) \equiv \neg ( \{ p \in \mathbb{X} : \phi(p) \} = \mathbb{X} )\)</span></li>
</ul></li>
</ul></li>
<li>Order of quantifiers matters.</li>
<li><p>Order of quantifiers matters.</p></li>
</ul>
</ul>
<h2id="relations">Relations</h2>
<h2id="relations">Relations</h2>
<ul>
<ul>
<li><p>Cartesian product:</p>
<li>Cartesian product:
<ul>
<ul>
<li>$A \times B = { (p, q) : p \in A \land q \in B }$</li>
<li><spanclass="math inline">\(A \times B = \{ (p, q) : p \in A \land q \in B \}\)</span></li>
</ul></li>
</ul></li>
<li>Def: A relation <spanclass="math inline">\(R\)</span> on a set <spanclass="math inline">\(\mathbb X\)</span> is a subset of <spanclass="math inline">\(\mathbb X \times \mathbb X\)</span> (<spanclass="math inline">\(R \subseteq \mathbb X \times \mathbb X\)</span>)</li>
<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>
<li>Graph of a function <spanclass="math inline">\(f()\)</span>: <spanclass="math inline">\(\{ (x, f(x) : x \in Dom(f) \}\)</span></li>
<li>Properties of:
<li><p>Graph of a function $f()$: ${ (x, f(x) : x \in Dom(f) }$</p></li>
<li><p>Properties of:</p>
<ul>
<ul>
<li>Reflexivity: $(\forall x \in \mathbb X ) (x, x) \in R \equiv (\forall x \in \mathbb X) x R x$</li>
<li>Reflexivity: <spanclass="math inline">\((\forall x \in \mathbb X ) (x, x) \in R \equiv (\forall x \in \mathbb X) x R x\)</span></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>Symmetricity: <spanclass="math inline">\([ (\forall x, y \in \mathbb X) (x, y) \in R \implies (y, x) \in R) ] \equiv [ (\forall x, y \in \mathbb X) ( x R y \implies y R x) ]\)</span></li>
<li>Transitivity: $(\forall x, y, z \in \mathbb X) (x R y \land y R z \implies x R z)$</li>
<li>Transitivity: <spanclass="math inline">\((\forall x, y, z \in \mathbb X) (x R y \land y R z \implies x R z)\)</span></li>
<li>Antisymmetricity: $(\forall x, y \in \mathbb X) (x R y \land y R x \implies x = y)$</li>
<li>Antisymmetricity: <spanclass="math inline">\((\forall x, y \in \mathbb X) (x R y \land y R x \implies x = y)\)</span></li>
</ul></li>
</ul></li>
<li>Equivalence relations:
<li><p>Equivalence relations:</p>
<ul>
<ul>
<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>Def: <spanclass="math inline">\(R \subseteq \mathbb X \times \mathbb X\)</span> is said to be an equivalence relation <em>iff</em><spanclass="math inline">\(R\)</span> is reflexive, symmetric and transitive.</li>
<li>Congruence modulo n: $p R q \equiv n | p - q$</li>
<li>Congruence modulo n: <spanclass="math inline">\(p R q \equiv n | p - q\)</span></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 $[x]_R = { y \in \mathbb X : x R y }$
<li>Def R - and equivalence relation of <spanclass="math inline">\(\mathbb X\)</span>: The <em>equivalence class</em> of an element <spanclass="math inline">\(x \in \mathbb X\)</span> is the set <spanclass="math inline">\([x]_R = \{ y \in \mathbb X : x R y \}\)</span>
<ul>
<ul>
<li>Every $x \in \mathbb X$ belongs to the equivalence class of some element $a$.</li>
<li>Every <spanclass="math inline">\(x \in \mathbb X\)</span> belongs to the equivalence class of some element <spanclass="math inline">\(a\)</span>.</li>
<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>
<li>A partition is a set containing subsets of some set <spanclass="math inline">\(\mathbb X\)</span> such that their collective symmetric difference equals <spanclass="math inline">\(\mathbb X\)</span>. A partition of is a set <spanclass="math inline">\(\{ A_i: i \in \mathbb I \land A_i \subseteq \mathbb X \}\)</span> such that:
<li>$(\forall i, j \in \mathbb I) (i \neq j \implies A_i \cap A_j = \emptyset)$</li>
<li><spanclass="math inline">\((\forall i, j \in \mathbb I) (i \neq j \implies A_i \cap A_j = \emptyset)\)</span></li>
</ul></li>
</ul></li>
<li><spanclass="math inline">\(\{ A_i \}_{i \in \mathbb I}\)</span> is a partition <em>iff</em> there exists an equivalence relation <spanclass="math inline">\(R\)</span> on <spanclass="math inline">\(\mathbb X\)</span> such that:
<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>
<ul>
<ul>
<li>$(\forall i \in \mathbb I) (\exists x \in \mathbb X) A_i = [x]_R$</li>
<li><spanclass="math inline">\((\forall i \in \mathbb I) (\exists x \in \mathbb X) A_i = [x]_R\)</span></li>
<li>The quotient set: <spanclass="math inline">\(\mathbb X / R = \{ [a] : a \in \mathbb X \}\)</span></li>
<li><p>The quotient set: $\mathbb X / R = { [a] : a \in \mathbb X }$</p></li>
</ul></li>
</ul></li>
</ul>
</ul>
<h2id="posets">Posets</h2>
<h2id="posets">Posets</h2>
<ul>
<ul>
<li><p>Partial orders</p>
<li>Partial orders
<ul>
<ul>
<li>$\mathbb X$ is a set, $R \subseteq \mathbb X \times \mathbb X$</li>
<li><spanclass="math inline">\(\mathbb X\)</span> is a set, <spanclass="math inline">\(R \subseteq \mathbb X \times \mathbb X\)</span></li>
<li>Def: <spanclass="math inline">\(R\)</span> is a partial order on <spanclass="math inline">\(\mathbb X\)</span> iff <spanclass="math inline">\(R\)</span> is:
<li><p>Def: $R$ is a partial order on $\mathbb X$ iff $R$ is:</p>
<ul>
<ul>
<li>Reflexive</li>
<li>Reflexive</li>
<li>Antisymmetric</li>
<li>Antisymmetric</li>
<li>Transitive</li>
<li>Transitive</li>
</ul></li>
</ul></li>
<li>Def: <spanclass="math inline">\(m \in \mathbb X\)</span> is said to be:
<li><p>Def: $m \in \mathbb X$ is said to be:</p>
<ul>
<ul>
<li>maximal element in $(\mathbb X, \preccurlyeq)$ iff $(\forall a \in \mathbb X) m \preccurlyeq a \implies m = a$</li>
<li>maximal element in <spanclass="math inline">\((\mathbb X, \preccurlyeq)\)</span> iff <spanclass="math inline">\((\forall a \in \mathbb X) m \preccurlyeq a \implies m = a\)</span></li>
<li>largest iff $(\forall a \in \mathbb X) (a \preccurlyeq m)$</li>
<li>largest iff <spanclass="math inline">\((\forall a \in \mathbb X) (a \preccurlyeq m)\)</span></li>
<li>minimal iff $(\forall a \in \mathbb X) (a \preccurlyeq m \implies a = m)$</li>
<li>minimal iff <spanclass="math inline">\((\forall a \in \mathbb X) (a \preccurlyeq m \implies a = m)\)</span></li>
<li>smallest iff $(\forall a \in \mathbb X) (m \preccurlyeq a)$</li>
<li>Def: A partial order <spanclass="math inline">\(R\)</span> on <spanclass="math inline">\(\mathbb X\)</span> is said to be <em>“total”</em> iff <spanclass="math inline">\((\forall x, y \in \mathbb X) (x R y \lor y R x)\)</span></li>
<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>
<li>Def: A subset <spanclass="math inline">\(B\)</span> of <spanclass="math inline">\(\mathbb X\)</span> is called a chain <em>“chain”</em> iff <spanclass="math inline">\(B\)</span> is totally ordered by <spanclass="math inline">\(R\)</span>
<li><p>Def: A subset $B$ of $\mathbb X$ is called a chain <em>“chain”</em> iff $B$ is totally ordered by $R$</p>
<ul>
<ul>
<li>$C(\mathbb X)$ - the set of all chains in $(\mathbb X, R)$</li>
<li><spanclass="math inline">\(C(\mathbb X)\)</span> - the set of all chains in <spanclass="math inline">\((\mathbb X, R)\)</span></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>A chain <spanclass="math inline">\(D\)</span> in <spanclass="math inline">\((\mathbb X, R)\)</span> is called a maximal chain iff <spanclass="math inline">\(D\)</span> is a maximal element in <spanclass="math inline">\((C(\mathbb X), R)\)</span></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><spanclass="math inline">\(K \subseteq \mathbb X\)</span> is called an antichain in <spanclass="math inline">\((\mathbb X, R)\)</span> iff <spanclass="math inline">\((\forall p, q \in K) (p R q \lor q R p \implies p = q)\)</span></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>
<li>Def: <spanclass="math inline">\(R\)</span> is a partial order on <spanclass="math inline">\(\mathbb X\)</span>, <spanclass="math inline">\(R\)</span> is called a <em>well</em> order iff <spanclass="math inline">\(R\)</span> is a total order on <spanclass="math inline">\(X\)</span> and every nonempty subset <spanclass="math inline">\(A\)</span> of <spanclass="math inline">\(\mathbb X\)</span> has the smallest element</li>
</ul></li>
</ul></li>
</ul></li>
</ul></li>
</ul>
</ul>
<h2id="induction">Induction</h2>
<h2id="induction">Induction</h2>
<ul>
<ul>
<li>If $\phi$ is a propositional function defined on $\mathbb N$, if:
<li>If <spanclass="math inline">\(\phi\)</span> is a propositional function defined on <spanclass="math inline">\(\mathbb N\)</span>, if:
<li>$(\forall n \geq 1) \phi(n) \implies \phi(n+1)$</li>
<li><spanclass="math inline">\((\forall n \geq 1) \phi(n) \implies \phi(n+1)\)</span></li>
<li>$(\forall k \geq 1) \phi(k)$</li>
<li><spanclass="math inline">\((\forall k \geq 1) \phi(k)\)</span></li>
</ul></li>
</ul></li>
</ul>
</ul>
<h2id="functions">Functions</h2>
<h2id="functions">Functions</h2>
<ul>
<ul>
<li>$f: \mathbb X \to \mathbb Y$</li>
<li><spanclass="math inline">\(f: \mathbb X \to \mathbb Y\)</span></li>
<li>Def: <spanclass="math inline">\(f \subseteq \mathbb X \times \mathbb Y\)</span> is said to be a function if:
<li><p>Def: $f \subseteq \mathbb X \times \mathbb Y$ is said to be a function if:</p>
<ul>
<ul>
<li>$(\forall x \in \mathbb X)(\exists y \in \mathbb Y) (x, y) \in f(y = f(x))$</li>
<li><spanclass="math inline">\((\forall x \in \mathbb X)(\exists y \in \mathbb Y) (x, y) \in f(y = f(x))\)</span></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>
<li><spanclass="math inline">\((\forall a \in \mathbb X)(\forall p, q \in \mathbb Y)((a, p) \in f \land (a, q) \in f \implies p = q)\)</span></li>
</ul></li>
</ul></li>
<li>Types of functions <spanclass="math inline">\(f: \mathbb X \to \mathbb Y\)</span>:
<li><p>Types of functions $f: \mathbb X \to \mathbb Y$:</p>
<ul>
<ul>
<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><spanclass="math inline">\(f\)</span> is said to be an injection ( 1 to 1 function) iff <spanclass="math inline">\((\forall x_1, x_2 \in \mathbb X) x_1 \neq x_2 \implies f(x_1) \neq f(x_2)\)</span></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><spanclass="math inline">\(f\)</span> is said to be a surjection (onto function) iff <spanclass="math inline">\((\forall y \in \mathbb Y)(\exists x \in \mathbb X) f(x) = y\)</span></li>
<li>If $f^{-1}$ is a function from $\mathbb Y \to \mathbb X$ then $f^{-1}$ is called the inverse function for $f$
<li>If <spanclass="math inline">\(f^{-1}\)</span> is a function from <spanclass="math inline">\(\mathbb Y \to \mathbb X\)</span> then <spanclass="math inline">\(f^{-1}\)</span> is called the inverse function for <spanclass="math inline">\(f\)</span>
<ul>
<ul>
<li>Fact: $f^{-1}$ is a function iff $f$ is a <em>bijection</em> (1 to 1 and onto)</li>
<li>Fact: <spanclass="math inline">\(f^{-1}\)</span> is a function iff <spanclass="math inline">\(f\)</span> is a <em>bijection</em> (1 to 1 and onto)</li>
</ul></li>
</ul></li>
</ul></li>
</ul></li>
<li>For some set <spanclass="math inline">\(\mathbb A\)</span> the image of <spanclass="math inline">\(\mathbb A\)</span> by <spanclass="math inline">\(f\)</span> is <spanclass="math inline">\(f(\mathbb A) = \{ f(x) : x \in \mathbb A \}\)</span>. We can also define the inverse of an image even when the function itself isn’t invertible: <spanclass="math inline">\(f^{-1}(\mathbb A)\)</span></li>
<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>
Mathematics without infinitely small, continuous mathematical objects. The mathematics of finite sets. Propositional calculus 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.
Mathematics without infinitely small, continuous mathematical objects. The mathematics of finite sets. Propositional calculus 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. (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.
$2=7$ 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.
<pubDate>Wed, 20 Nov 2019 00:00:00 +0000</pubDate>
<pubDate>Wed, 20 Nov 2019 00:00:00 +0000</pubDate>
<guid>https://abdulocra.cy/posts/eidma/</guid>
<guid>https://abdulocra.cy/posts/eidma/</guid>
<description>Mathematics without infinitely small, continuous mathematical objects. The mathematics of finite sets. Propositional calculus 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.
<description>Mathematics without infinitely small, continuous mathematical objects. The mathematics of finite sets. Propositional calculus 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. \(2=7\) 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.</description>
$2=7$ 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.</description>
Mathematics without infinitely small, continuous mathematical objects. The mathematics of finite sets. Propositional calculus 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.
Mathematics without infinitely small, continuous mathematical objects. The mathematics of finite sets. Propositional calculus 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. (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.
$2=7$ 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.
<pubDate>Wed, 20 Nov 2019 00:00:00 +0000</pubDate>
<pubDate>Wed, 20 Nov 2019 00:00:00 +0000</pubDate>
<guid>https://abdulocra.cy/posts/eidma/</guid>
<guid>https://abdulocra.cy/posts/eidma/</guid>
<description>Mathematics without infinitely small, continuous mathematical objects. The mathematics of finite sets. Propositional calculus 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.
<description>Mathematics without infinitely small, continuous mathematical objects. The mathematics of finite sets. Propositional calculus 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. \(2=7\) 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.</description>
$2=7$ 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.</description>
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