Algebraic manipulation of "true" and "false"

Boolean algebra

Boolean algebra

In <a href="/en/Mathematics" title="Mathematics" class="wl">mathematics</a> and <a href="/en/Mathematical_logic" title="Mathematical logic" class="wl">mathematical logic</a>, Boolean algebra is a branch of <a href="/en/Algebra" title="Algebra" class="wl">algebra</a>. It differs from <a href="/en/Elementary_algebra" title="Elementary algebra" class="wl">elementary algebra</a> in two ways. First, the values of the <a href="/en/Variable_(mathematics)" title="Variable (mathematics)" class="wl">variables</a> are the <a href="/en/Truth_value" title="Truth value" class="wl">truth values</a> true and false, usually denoted 1 and 0, whereas in elementary algebra the values of the variables are numbers. Second, Boolean algebra uses <a href="/en/Logical_connective" title="Logical connective" class="wl">logical operators</a> such as <a href="/en/Logical_conjunction" title="Logical conjunction" class="wl">conjunction</a> (and) denoted as ∧, <a href="/en/Logical_disjunction" title="Logical disjunction" class="wl">disjunction</a> (or) denoted as ∨, and the <a href="/en/Negation" title="Negation" class="wl">negation</a> (not) denoted as ¬. Elementary algebra, on the other hand, uses arithmetic operators such as addition, multiplication, subtraction and division. Boolean algebra is therefore a formal way of describing <a href="/en/Logical_operation" title="Logical operation" class="mw-redirect wl">logical operations</a>, in the same way that elementary algebra describes numerical operations.

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Boolean algebra was introduced by <a href="/en/George_Boole" title="George Boole" class="wl">George Boole</a> in his first book The Mathematical Analysis of Logic<a href="#cite_note-Boole_2011-1" style="counter-reset: mw-Ref 1;">[1]</a> (1847), and set forth more fully in his <a href="/en/The_Laws_of_Thought" title="The Laws of Thought" class="wl">An Investigation of the Laws of Thought</a> (1854).<a href="#cite_note-Boole_1854-2" style="counter-reset: mw-Ref 2;">[2]</a>
According to <a href="/en/Edward_Vermilye_Huntington" title="Edward Vermilye Huntington" class="wl">Huntington</a>, the term "Boolean algebra" was first suggested by <a href="/en/Henry_M._Sheffer" title="Henry M. Sheffer" class="wl">Henry M. Sheffer</a> in 1913,<a href="#cite_note-Huntington_1933-3" style="counter-reset: mw-Ref 3;">[3]</a> although <a href="/en/Charles_Sanders_Peirce" title="Charles Sanders Peirce" class="wl">Charles Sanders Peirce</a> gave the title "A Boolean Algebra with One Constant" to the first chapter of his "The Simplest Mathematics" in 1880.<a href="#cite_note-Peirce_1931-4" style="counter-reset: mw-Ref 4;">[4]</a>
Boolean algebra has been fundamental in the development of <a href="/en/Digital_electronics" title="Digital electronics" class="wl">digital electronics</a>, and is provided for in all modern <a href="/en/Programming_language" title="Programming language" class="wl">programming languages</a>. It is also used in <a href="/en/Set_theory" title="Set theory" class="wl">set theory</a> and <a href="/en/Statistics" title="Statistics" class="wl">statistics</a>.<a href="#cite_note-Givant-Halmos_2009-5" style="counter-reset: mw-Ref 5;">[5]</a>



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 elementary algebra the values of the variables are numbers. Second, Boolean algebra uses logical operators such as conjunction denoted as ∧, disjunction denoted as ∨, and the negation denoted as ¬. Elementary algebra, on the other hand, uses arithmetic operators such as addition, multiplication, subtraction and division. Boolean algebra is therefore a formal way of describing logical operations, in the same way that elementary algebra describes numerical operations. Boolean algebra was introduced by George Boole in his first book The Mathematical Analysis of Logic 