In

A demonstration of the Distributive Law

for integer arithmetic (from

mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the distributive property of binary operation
In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two.
More specifically, an internal binary o ...

s generalizes the distributive law, which asserts that the equality
$$x\; \backslash cdot\; (y\; +\; z)\; =\; x\; \backslash cdot\; y\; +\; x\; \backslash cdot\; z$$
is always true in elementary algebra
Elementary algebra encompasses the basic concepts of algebra. It is often contrasted with arithmetic: arithmetic deals with specified numbers, whilst algebra introduces variables (quantities without fixed values).
This use of variables entail ...

.
For example, in elementary arithmetic, one has
$$2\; \backslash cdot\; (1\; +\; 3)\; =\; (2\; \backslash cdot\; 1)\; +\; (2\; \backslash cdot\; 3).$$
One says that multiplication
Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being addi ...

''distributes'' over addition
Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or '' sum'' of ...

.
This basic property of numbers is part of the definition of most algebraic structure
In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set o ...

s that have two operations called addition and multiplication, such as complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...

s, polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exam ...

s, matrices, rings
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...

, and fields
Fields may refer to:
Music
*Fields (band), an indie rock band formed in 2006
*Fields (progressive rock band), a progressive rock band formed in 1971
* ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010)
* "Fields", a song by ...

. It is also encountered in Boolean algebra
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas in ...

and mathematical logic
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal ...

, where each of the logical and (denoted $\backslash ,\backslash land\backslash ,$) and the logical or
In logic, disjunction is a logical connective typically notated as \lor and read aloud as "or". For instance, the English language sentence "it is raining or it is snowing" can be represented in logic using the disjunctive formula R \lor S ...

(denoted $\backslash ,\backslash lor\backslash ,$) distributes over the other.
Definition

Given aset
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...

$S$ and two binary operators $\backslash ,*\backslash ,$ and $\backslash ,+\backslash ,$ on $S,$
*the operation $\backslash ,*\backslash ,$ is over (or with respect to) $\backslash ,+\backslash ,$ if, given any elements $x,\; y,\; \backslash text\; z$ of $S,$
$$x\; *\; (y\; +\; z)\; =\; (x\; *\; y)\; +\; (x\; *\; z);$$
*the operation $\backslash ,*\backslash ,$ is over $\backslash ,+\backslash ,$ if, given any elements $x,\; y,\; \backslash text\; z$ of $S,$
$$(y\; +\; z)\; *\; x\; =\; (y\; *\; x)\; +\; (z\; *\; x);$$
*and the operation $\backslash ,*\backslash ,$ is over $\backslash ,+\backslash ,$ if it is left- and right-distributive.
When $\backslash ,*\backslash ,$ is commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...

, the three conditions above are logically equivalent.
Meaning

The operators used for examples in this section are those of the usualaddition
Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or '' sum'' of ...

$\backslash ,+\backslash ,$ and multiplication
Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being addi ...

$\backslash ,\backslash cdot.\backslash ,$
If the operation denoted $\backslash cdot$ is not commutative, there is a distinction between left-distributivity and right-distributivity:
$$a\; \backslash cdot\; \backslash left(\; b\; \backslash pm\; c\; \backslash right)\; =\; a\; \backslash cdot\; b\; \backslash pm\; a\; \backslash cdot\; c\; \backslash qquad\; \backslash text$$
$$(a\; \backslash pm\; b)\; \backslash cdot\; c\; =\; a\; \backslash cdot\; c\; \backslash pm\; b\; \backslash cdot\; c\; \backslash qquad\; \backslash text.$$
In either case, the distributive property can be described in words as:
To multiply a sum (or difference) by a factor, each summand (or minuend and subtrahend) is multiplied by this factor and the resulting products are added (or subtracted).
If the operation outside the parentheses (in this case, the multiplication) is commutative, then left-distributivity implies right-distributivity and vice versa, and one talks simply of .
One example of an operation that is "only" right-distributive is division, which is not commutative:
$$(a\; \backslash pm\; b)\; \backslash div\; c\; =\; a\; \backslash div\; c\; \backslash pm\; b\; \backslash div\; c.$$
In this case, left-distributivity does not apply:
$$a\; \backslash div(b\; \backslash pm\; c)\; \backslash neq\; a\; \backslash div\; b\; \backslash pm\; a\; \backslash div\; c$$
The distributive laws are among the axioms for rings
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...

(like the ring of integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language o ...

s) and fields
Fields may refer to:
Music
*Fields (band), an indie rock band formed in 2006
*Fields (progressive rock band), a progressive rock band formed in 1971
* ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010)
* "Fields", a song by ...

(like the field of rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rati ...

s). Here multiplication is distributive over addition, but addition is not distributive over multiplication. Examples of structures with two operations that are each distributive over the other are Boolean algebras such as the algebra of sets
In mathematics, the algebra of sets, not to be confused with the mathematical structure of ''an'' algebra of sets, defines the properties and laws of sets, the set-theoretic operations of union, intersection, and complementation and the re ...

or the switching algebra.
Multiplying sums can be put into words as follows: When a sum is multiplied by a sum, multiply each summand of a sum with each summand of the other sum (keeping track of signs) then add up all of the resulting products.
Examples

Real numbers

In the following examples, the use of the distributive law on the set of real numbers $\backslash R$ is illustrated. When multiplication is mentioned in elementary mathematics, it usually refers to this kind of multiplication. From the point of view of algebra, the real numbers form a field, which ensures the validity of the distributive law.Matrices

The distributive law is valid formatrix multiplication
In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the s ...

. More precisely,
$$(A\; +\; B)\; \backslash cdot\; C\; =\; A\; \backslash cdot\; C\; +\; B\; \backslash cdot\; C$$
for all $l\; \backslash times\; m$-matrices $A,\; B$ and $m\; \backslash times\; n$-matrices $C,$ as well as
$$A\; \backslash cdot\; (B\; +\; C)\; =\; A\; \backslash cdot\; B\; +\; A\; \backslash cdot\; C$$
for all $l\; \backslash times\; m$-matrices $A$ and $m\; \backslash times\; n$-matrices $B,\; C.$
Because the commutative property does not hold for matrix multiplication, the second law does not follow from the first law. In this case, they are two different laws.
Other examples

*Multiplication
Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being addi ...

of ordinal number
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets.
A finite set can be enumerated by successively labeling each element with the least n ...

s, in contrast, is only left-distributive, not right-distributive.
* The cross product
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is ...

is left- and right-distributive over vector addition
In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors ...

, though not commutative.
* The union of sets is distributive over intersection, and intersection is distributive over union.
* Logical disjunction
In logic, disjunction is a logical connective typically notated as \lor and read aloud as "or". For instance, the English language sentence "it is raining or it is snowing" can be represented in logic using the disjunctive formula R \lor S ...

("or") is distributive over logical conjunction
In logic, mathematics and linguistics, And (\wedge) is the truth-functional operator of logical conjunction; the ''and'' of a set of operands is true if and only if ''all'' of its operands are true. The logical connective that represents this ...

("and"), and vice versa.
* For real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...

s (and for any totally ordered set), the maximum operation is distributive over the minimum operation, and vice versa: $$\backslash max(a,\; \backslash min(b,\; c))\; =\; \backslash min(\backslash max(a,\; b),\; \backslash max(a,\; c))\; \backslash quad\; \backslash text\; \backslash quad\; \backslash min(a,\; \backslash max(b,\; c))\; =\; \backslash max(\backslash min(a,\; b),\; \backslash min(a,\; c)).$$
* For integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language o ...

s, the greatest common divisor
In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers ''x'', ''y'', the greatest common divisor of ''x'' and ''y'' is ...

is distributive over the least common multiple
In arithmetic and number theory, the least common multiple, lowest common multiple, or smallest common multiple of two integers ''a'' and ''b'', usually denoted by lcm(''a'', ''b''), is the smallest positive integer that is divisible by bo ...

, and vice versa: $$\backslash gcd(a,\; \backslash operatorname(b,\; c))\; =\; \backslash operatorname(\backslash gcd(a,\; b),\; \backslash gcd(a,\; c))\; \backslash quad\; \backslash text\; \backslash quad\; \backslash operatorname(a,\; \backslash gcd(b,\; c))\; =\; \backslash gcd(\backslash operatorname(a,\; b),\; \backslash operatorname(a,\; c)).$$
* For real numbers, addition distributes over the maximum operation, and also over the minimum operation: $$a\; +\; \backslash max(b,\; c)\; =\; \backslash max(a\; +\; b,\; a\; +\; c)\; \backslash quad\; \backslash text\; \backslash quad\; a\; +\; \backslash min(b,\; c)\; =\; \backslash min(a\; +\; b,\; a\; +\; c).$$
* For binomial
Binomial may refer to:
In mathematics
* Binomial (polynomial), a polynomial with two terms
*Binomial coefficient, numbers appearing in the expansions of powers of binomials
* Binomial QMF, a perfect-reconstruction orthogonal wavelet decomposition ...

multiplication, distribution is sometimes referred to as the FOIL Method (First terms $a\; c,$ Outer $a\; d,$ Inner $b\; c,$ and Last $b\; d$) such as: $(a\; +\; b)\; \backslash cdot\; (c\; +\; d)\; =\; a\; c\; +\; a\; d\; +\; b\; c\; +\; b\; d.$
* In all semirings, including the complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...

s, the quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...

s, polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exam ...

s, and matrices, multiplication distributes over addition: $u\; (v\; +\; w)\; =\; u\; v\; +\; u\; w,\; (u\; +\; v)w\; =\; u\; w\; +\; v\; w.$
* In all algebras over a field, including the octonion
In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions have ...

s and other non-associative algebras, multiplication distributes over addition.
Propositional logic

Rule of replacement

In standard truth-functional propositional logic, in logical proofs uses two valid rules of replacement to expand individual occurrences of certainlogical connective
In logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is a logical constant. They can be used to connect logical formulas. For instance in the syntax of propositional logic, the binary ...

s, within some formula
In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betwe ...

, into separate applications of those connectives across subformulas of the given formula. The rules are
$$(P\; \backslash land\; (Q\; \backslash lor\; R))\; \backslash Leftrightarrow\; ((P\; \backslash land\; Q)\; \backslash lor\; (P\; \backslash land\; R))\; \backslash qquad\; \backslash text\; \backslash qquad\; (P\; \backslash lor\; (Q\; \backslash land\; R))\; \backslash Leftrightarrow\; ((P\; \backslash lor\; Q)\; \backslash land\; (P\; \backslash lor\; R))$$
where "$\backslash Leftrightarrow$", also written $\backslash ,\backslash equiv,\backslash ,$ is a metalogical symbol
A symbol is a mark, sign, or word that indicates, signifies, or is understood as representing an idea, object, or relationship. Symbols allow people to go beyond what is known or seen by creating linkages between otherwise very different ...

representing "can be replaced in a proof with" or "is logically equivalent to".
Truth functional connectives

is a property of some logical connectives of truth-functional propositional logic. The following logical equivalences demonstrate that distributivity is a property of particular connectives. The following are truth-functional tautologies. ;Double distribution: $$\backslash begin\; \&((P\; \backslash land\; Q)\; \&\&\backslash ;\backslash lor\; (R\; \backslash land\; S))\; \&\&\backslash ;\backslash Leftrightarrow\backslash ;\&\&\; (((P\; \backslash lor\; R)\; \backslash land\; (P\; \backslash lor\; S))\; \&\&\backslash ;\backslash land\; ((Q\; \backslash lor\; R)\; \backslash land\; (Q\; \backslash lor\; S)))\; \&\&\; \backslash \backslash \; \&((P\; \backslash lor\; Q)\; \&\&\backslash ;\backslash land\; (R\; \backslash lor\; S))\; \&\&\backslash ;\backslash Leftrightarrow\backslash ;\&\&\; (((P\; \backslash land\; R)\; \backslash lor\; (P\; \backslash land\; S))\; \&\&\backslash ;\backslash lor\; ((Q\; \backslash land\; R)\; \backslash lor\; (Q\; \backslash land\; S)))\; \&\&\; \backslash \backslash \; \backslash end$$Distributivity and rounding

In approximate arithmetic, such asfloating-point arithmetic
In computing, floating-point arithmetic (FP) is arithmetic that represents real numbers approximately, using an integer with a fixed precision, called the significand, scaled by an integer exponent of a fixed base. For example, 12.345 can be r ...

, the distributive property of multiplication (and division) over addition may fail because of the limitations of arithmetic precision
Significant figures (also known as the significant digits, ''precision'' or ''resolution'') of a number in positional notation are digits in the number that are reliable and necessary to indicate the quantity of something.
If a number expres ...

. For example, the identity $1/3\; +\; 1/3\; +\; 1/3\; =\; (1\; +\; 1\; +\; 1)\; /\; 3$ fails in decimal arithmetic, regardless of the number of significant digits. Methods such as banker's rounding may help in some cases, as may increasing the precision used, but ultimately some calculation errors are inevitable.
In rings and other structures

Distributivity is most commonly found in semirings, notably the particular cases of rings and distributive lattices. A semiring has two binary operations, commonly denoted $\backslash ,+\backslash ,$ and $\backslash ,*,$ and requires that $\backslash ,*\backslash ,$ must distribute over $\backslash ,+.$ A ring is a semiring with additive inverses. A lattice is another kind ofalgebraic structure
In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set o ...

with two binary operations, $\backslash ,\backslash land\; \backslash text\; \backslash lor.$
If either of these operations distributes over the other (say $\backslash ,\backslash land\backslash ,$ distributes over $\backslash ,\backslash lor$), then the reverse also holds ($\backslash ,\backslash lor\backslash ,$ distributes over $\backslash ,\backslash land\backslash ,$), and the lattice is called distributive. See also .
A Boolean algebra
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas in ...

can be interpreted either as a special kind of ring (a Boolean ring) or a special kind of distributive lattice (a Boolean lattice
In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties of both set operations and logic operations. A Boolean algebra can be seen as a gen ...

). Each interpretation is responsible for different distributive laws in the Boolean algebra.
Similar structures without distributive laws are near-ring In mathematics, a near-ring (also near ring or nearring) is an algebraic structure similar to a ring but satisfying fewer axioms. Near-rings arise naturally from functions on groups.
Definition
A set ''N'' together with two binary operations ...

s and near-fields instead of rings and division rings. The operations are usually defined to be distributive on the right but not on the left.
Generalizations

In several mathematical areas, generalized distributivity laws are considered. This may involve the weakening of the above conditions or the extension to infinitary operations. Especially in order theory one finds numerous important variants of distributivity, some of which include infinitary operations, such as the infinite distributive law; others being defined in the presence of only binary operation, such as the according definitions and their relations are given in the article distributivity (order theory). This also includes the notion of a completely distributive lattice. In the presence of an ordering relation, one can also weaken the above equalities by replacing $\backslash ,=\backslash ,$ by either $\backslash ,\backslash leq\backslash ,$ or $\backslash ,\backslash geq.$ Naturally, this will lead to meaningful concepts only in some situations. An application of this principle is the notion of sub-distributivity as explained in the article oninterval arithmetic
Interval arithmetic (also known as interval mathematics, interval analysis, or interval computation) is a mathematical technique used to put bounds on rounding errors and measurement errors in mathematical computation. Numerical methods usi ...

.
In category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...

, if $(S,\; \backslash mu,\; \backslash nu)$ and $\backslash left(S^,\; \backslash mu^,\; \backslash nu^\backslash right)$ are monads on a category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
*Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce)
...

$C,$ a distributive law $S\; .\; S^\; \backslash to\; S^\; .\; S$ is a natural transformation $\backslash lambda\; :\; S\; .\; S^\; \backslash to\; S^\; .\; S$ such that $\backslash left(S^,\; \backslash lambda\backslash right)$ is a lax map of monads $S\; \backslash to\; S$ and $(S,\; \backslash lambda)$ is a colax map of monads $S^\; \backslash to\; S^.$ This is exactly the data needed to define a monad structure on $S^\; .\; S$: the multiplication map is $S^\; \backslash mu\; .\; \backslash mu^\; S^2\; .\; S^\; \backslash lambda\; S$ and the unit map is $\backslash eta^\; S\; .\; \backslash eta.$ See: distributive law between monads.
A generalized distributive law has also been proposed in the area of information theory
Information theory is the scientific study of the quantification, storage, and communication of information. The field was originally established by the works of Harry Nyquist and Ralph Hartley, in the 1920s, and Claude Shannon in the 194 ...

.
Antidistributivity

The ubiquitous identity that relates inverses to the binary operation in any group, namely $(x\; y)^\; =\; y^\; x^,$ which is taken as an axiom in the more general context of a semigroup with involution, has sometimes been called an antidistributive property (of inversion as aunary operation
In mathematics, an unary operation is an operation with only one operand, i.e. a single input. This is in contrast to binary operations, which use two operands. An example is any function , where is a set. The function is a unary operation ...

).
In the context of a near-ring In mathematics, a near-ring (also near ring or nearring) is an algebraic structure similar to a ring but satisfying fewer axioms. Near-rings arise naturally from functions on groups.
Definition
A set ''N'' together with two binary operations ...

, which removes the commutativity of the additively written group and assumes only one-sided distributivity, one can speak of (two-sided) distributive elements but also of antidistributive elements. The latter reverse the order of (the non-commutative) addition; assuming a left-nearring (i.e. one which all elements distribute when multiplied on the left), then an antidistributive element $a$ reverses the order of addition when multiplied to the right: $(x\; +\; y)\; a\; =\; y\; a\; +\; x\; a.$
In the study of propositional logic and Boolean algebra
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas in ...

, the term antidistributive law is sometimes used to denote the interchange between conjunction and disjunction when implication factors over them:
$$(a\; \backslash lor\; b)\; \backslash Rightarrow\; c\; \backslash equiv\; (a\; \backslash Rightarrow\; c)\; \backslash land\; (b\; \backslash Rightarrow\; c)$$
$$(a\; \backslash land\; b)\; \backslash Rightarrow\; c\; \backslash equiv\; (a\; \backslash Rightarrow\; c)\; \backslash lor\; (b\; \backslash Rightarrow\; c).$$
These two tautologies are a direct consequence of the duality in De Morgan's laws.
Notes

External links

{{Wiktionary, distributivityA demonstration of the Distributive Law

for integer arithmetic (from

cut-the-knot
Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet-born Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and M ...

)
Properties of binary operations
Elementary algebra
Rules of inference
Theorems in propositional logic