...n which order is significant; $(a,b)$ is not the same as $(b,a)$ unless $a=b$. Equality between ordered pairs is defined by
(a,b) = (c,d) \ \Leftrightarrow \ a=c \wedge b=d \ .
662 bytes (112 words) - 11:56, 23 November 2023
...$AB$. The expressions $A$ and $B$ are called the conjunctive terms of $A\&B$. The normal usage of the conjunction in mathematical logic corresponds to
|$A$||$B$||$A\&B$
623 bytes (106 words) - 07:28, 12 August 2014
...B$ has the property that for every $a \in A$ there is $b \in B$ such that $b \le a$.
451 bytes (77 words) - 20:25, 20 November 2023
$$I_x(a,b)=\frac1{B(a,b)}\int\limits_0^xt^{a-1}(1-t)^{b-1}dt,$$
$$0\leq x\leq1,\quad a>0,\quad b>0,$$
2 KB (393 words) - 15:50, 31 March 2017
$$\max\{A,B\} = \begin{cases} A & \mathrm{ if\ } A\ge B\\
B & \mathrm{ otherwise }
240 bytes (35 words) - 12:19, 28 January 2012
...triangle in the Euclidean plane with sides $a,b,c$ and opposite angles $A,B,C$, respectively, the equalities
$$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2R$$
671 bytes (114 words) - 14:29, 19 March 2014
...ments $A$ and $B$ are called the disjunctive terms of the statement $A\lor B$. The meaning of the disjunction can be expressed by the following [[Truth
|$A$||$B$||$A\lor B$
573 bytes (98 words) - 07:30, 12 August 2014
* $|a|-|b|\leq |a+b| \leq |a| + |b|$,
* $|a|-|b|\leq |a-b| \leq |a| + |b|$,
752 bytes (129 words) - 16:51, 30 November 2014
...triple product''' or ''[[mixed product]]'' of vectors $\mathbf{a},\mathbf{b},\mathbf{c}$ is
...= \mathbf{a} \times \mathbf{b} \cdot \mathbf{c} = \mathbf{a} \cdot \mathbf{b} \times \mathbf{c}\ ;
1 KB (176 words) - 19:37, 6 December 2023
...ied to $b$. Then $G \mapsto g$ defines a map from $(A^B)^C \rightarrow A^{B \times C}$. These two correspondences are mutually inverse.
\mathfrak{a}^{\mathfrak{b}\mathfrak{c}} = (\mathfrak{a}^{\mathfrak{b}})^{\mathfrak{c}} \ .
2 KB (289 words) - 11:59, 23 November 2023
| style="text-align: center;" | '''A''' || '''All''' $A$ '''are''' $B$ ||
| style="text-align: center;" | '''I''' || '''Some''' $A$ '''are''' $B$ ||
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...be a continuous mapping and let $m < n$. Then there is an $x \in \partial B$ such that $\phi(x) = \phi(-x)$.
The original version (K. Borsuk, 1933) was for $B = D^n$, the $n$-dimensional ball in $\mathbf{R}^n$, $n = m$. The result is
1 KB (208 words) - 20:43, 19 January 2016
...ich contains the end points, i.e. consists of the points $x$, $a\leq x\leq b$.
...increasingly popular) alternative $]a,b[$ to denote the open interval $a<x<b$.
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...taken to be true if $A$ is true and $B$ is false, or if $A$ is false and $B$ is true. In the remaining cases it is taken to be false. Thus, the exclusi
\[A\mathbin{\dot\lor}B\Leftrightarrow(A\lor B)\mathbin\&\neg(A\land B).\]
500 bytes (90 words) - 13:49, 30 December 2018
...such that if $ a \leq b $ (where $ a,b \in A $), then $ \phi(a) \geq \phi(b) $. The dual concept to an antitone mapping is an [[isotone mapping]].
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...\phi : A \otimes_{k} B \to \Omega$ that maps the tensor product $a \otimes b$ to $ab \in \Omega$.
347 bytes (59 words) - 21:46, 22 October 2017
\mu (B)= \sup \{\mu(K): K\subset B, \mbox{ $K$ compact} \}\,
\mu (B)= \sup \{\mu(K): K\subset B, \mbox{ $K$ compact} \}\,
679 bytes (103 words) - 21:39, 20 June 2014
...ct of $A$ and $B$ can be defined without assuming in advance that $A$ and $B$ are subgroups of a certain group $C$.
857 bytes (151 words) - 21:01, 10 January 2017
...$, where $[a]$ is the class containing the element $a$, if $a\triangleleft b$. For examples and references see [[Partially ordered set]].
...me writers use the notation $a||b$ to indicate that neither $a\leq b$ nor $b\leq a$ holds.
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...tarrow B $ such that $ a < b $ in $ A $ implies $ \phi(a) > \phi(b) $ in $ B $ (and similarly for the inverse).
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...c$, that is, for each $a\in A$ there is at most one $b\in A$ such that $(a,b)\in R$. Thus, $R$ determines a function (perhaps not defined everywhere) on
...B$ between sets $A$ and $B$ such that $(a,b)\in R$ and $(a,c)\in R$ imply $b=c$.
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$$A\supset(\neg A\supset B).$$
...of intuitionistic propositional calculus is formulated as follows. Let $A,B,C$ be arbitrary formulas in the logical language considered. The axioms of
2 KB (271 words) - 21:33, 15 December 2015
...he first). Then the set of those elements of $A$ that are not elements of $B$ is called the difference of these sets.
The difference between two sets $A$ and $B$ is denoted by $A\setminus B$.
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a+b=b+a,\quad \text{ and } \quad ab=ba.
...e law of commutativity) if in the given algebraic system the identity $a*b=b*a$ holds.
427 bytes (66 words) - 22:17, 26 October 2014
|$A$||$B$||$A\downarrow B$
...junction $A\lor B$ is equivalent to $(A\downarrow B)\downarrow(A\downarrow B)$. This arrow was introduced by C. Peirce.
1 KB (194 words) - 12:27, 12 August 2014
$$A\supset(B\supset A),\quad(A\supset(B\supset C))\supset((A\supset B)\supset(A\supset C)),$$
$$A\&B\supset A,\quad A\&B\supset B,\quad A\supset(B\supset A\&B),$$
2 KB (246 words) - 08:00, 12 August 2014
''of an arithmetical fraction $a/b$''
...raction. The denominator of an algebraic fraction $A/B$ is the expression $B$ (see [[Fraction|Fraction]]).
340 bytes (49 words) - 09:49, 15 April 2014
...$a$ and $b$ and including the end points $a$ and $b$. It is denoted by $[a,b]$. See also [[Interval and segment|Interval and segment]].
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''of an arithmetic fraction $a/b$''
...is used to make up the fraction. The numerator of an algebraic fraction $A/B$ is the expression $A$ (cf. [[Fraction|Fraction]]).
238 bytes (42 words) - 09:50, 15 April 2014
...of $a$ and $b$. If this number $n$ can be chosen independently of $a$ and $b$, then $G$ is called an Engel group of finite class $n$. The class of Engel
...pringer (1982)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> B. Huppert, "Finite groups" , '''3''' , Springer (1982)</TD></TR></table>
1 KB (200 words) - 11:46, 26 April 2014
...ment $x\in P$ such that $a<x<b$: the [[Interval and segment|interval]] $[a,b]$ is an ''atomic'' or ''[[elementary interval]]''.
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''$B$-function, Euler $B$-function, Euler integral of the first kind''
B(p,q) = \int_0^1 x^{p-1}(1-x)^{q-1} \rd x.
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|$A$||$B$||$A|B$
...A|A$; the [[Disjunction|disjunction]] $A\lor B$ of two assertions $A$ and $B$ is expressed as:
2 KB (249 words) - 17:56, 29 November 2014
''of vectors $\mathbf{a},\mathbf{b},\mathbf{c}$''
...product]] of the vector $\mathbf{a}$ by the [[vector product]] of $\mathbf{b}$ and $\mathbf{c}$:
397 bytes (60 words) - 08:09, 14 January 2018
...cture(2)|Structure]]), it follows that $A$ is [[existentially closed]] in $B$.
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...tent to a subset of the other), then there is a bijection between $A$ and $B$ (they are [[equipotent sets]]).
...{b}$ and $\mathfrak{b} \le \mathfrak{a}$ implies $\mathfrak{a} = \mathfrak{b}$.
1,006 bytes (155 words) - 19:36, 17 November 2023
$$x=a\cos t,\quad y=\pm\sqrt{b^2-a^2\sin^2t},\quad z=a\sin t,\quad b\geq a,$$
...e $a$ and $b$ are the radii of the cylinders and $t$ is a parameter. If $a=b$, the bicylindrics is a pair of congruent ellipses.
379 bytes (73 words) - 09:40, 5 August 2014
''of a group $A$ by a group $B$''
...h element $a \in A$ corresponds an automorphism $\alpha_a \in \mathrm{Aut}(B)$, which is [[conjugation]] by the element $a$:
2 KB (359 words) - 16:54, 23 November 2023
...\,b\,\omega$ or $(a,b)\omega$, but more commonly in infix form as $a \star b$ where $\star$ is the operator symbol. Many arithmetic, algebraic and logi
A binary operation is ''partial'' if it is not defined on all pairs $(a,b) \in A \times A$ (as for example division by zero is not defined). Propert
1 KB (193 words) - 19:47, 13 November 2016
...er product]] of vectors (where the product $(a,b)$ of two vectors $a$ and $b$ is, in general, a complex number) that satisfies the following axioms:
1) $(a,b)=\overline{(b,a)}$;
1 KB (173 words) - 05:40, 20 April 2023
...ddition of numbers is commutative: $a+b=b+a$, and associative: $(a+b)+c=a+(b+c)$. The operation inverse to addition is called subtraction.
739 bytes (126 words) - 20:47, 16 March 2014
Two subextensions $A$ and $B$ of an extension $\def\O{\Omega}\O$ of $k$ are called linearly disjoint if
subalgebra generated by $A$ and $B$ in $\O$ is (isomorphic to) the
1 KB (259 words) - 22:07, 5 March 2012
...out remainder. In other words, a divisor of the integer $a$ is an integer $b$ such that, for a certain integer $c$, the equality $a=bc$ holds. A ''prop
A divisor of a polynomial $A(x)$ is a polynomial $B(x)$ that divides $A(x)$ without remainder (cf. [[Division|Division]]).
1 KB (209 words) - 08:06, 26 November 2023
Linear operators $B$ and $T$, of which $T$ is of general type and $B$ is bounded, and which are such that
...rator|Extension of an operator]]). The commutation relation is denoted by $B\cup T$ and satisfies the following rules:
1 KB (259 words) - 15:29, 14 February 2020
''of two elements $a$ and $b$ in a group with multiple operators''
$$-a-b+a+b.$$
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...is called the consequence. The precise meaning of the expression $A\supset B$ differs in the classical, constructive and other approaches to the semanti
!$A$||$B$||$A\supset B$
1 KB (166 words) - 16:47, 30 December 2018
''of three elements $a,b,c$ of a ring $R$''
It is denoted by $(a,b,c)$.
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...is an [[incidence system]] on a set $V$ of $v$ points and a family $B$ of $b$ blocks: subsets of points, each of size $k$. It is required that each poi
b \ge v \ .
563 bytes (91 words) - 19:38, 4 January 2016
...expressing the surface area $S$ of a triangle in terms of its sides $a$, $b$ and $c$:
$$S=\sqrt{p(p-a)(p-b)(p-c)},$$
527 bytes (83 words) - 18:02, 17 April 2023
...ven two sets $A$ and $B$, their symmetric difference, denoted by $A \Delta B$, is given by
...\setminus A) = (A \cup B) \setminus (A \cap B) = (A \cap B') \cup (A' \cap B)
2 KB (273 words) - 08:47, 29 April 2023