{"id":995,"date":"2011-02-27T17:16:07","date_gmt":"2011-02-27T16:16:07","guid":{"rendered":"http:\/\/kmr.dialectica.se\/wp\/?page_id=995"},"modified":"2011-02-27T17:16:07","modified_gmt":"2011-02-27T16:16:07","slug":"disambiguating-equality","status":"publish","type":"page","link":"https:\/\/kmr.placify.me\/wp\/research\/math-rehab\/mathematical-concepts\/disambiguation\/disambiguating-equality\/","title":{"rendered":"Disambiguating equality"},"content":{"rendered":"

This page is a sub-page of our page on Disambiguation<\/a>. <\/p>\n

\/\/\/\/\/\/\/<\/p>\n

Like the term ‘add’ (represented by the \\, + \\, <\/span> sign), the term ‘equal’ (represented by the \\, = \\,<\/span> sign) has many different meanings in mathematics. In fact, the \\, = \\, <\/span> sign can stand for at least five different types of equality:<\/p>\n

1) Identical<\/strong> (or algebraic<\/strong>) equality:<\/p>\n

Examples<\/strong>: \\, 3 + 5 = 8 \\, <\/span>, and \\, (x + y) (x - y) = x^2 - y^2 \\, <\/span>.<\/p>\n

Notation<\/em>: This type of equality (identical equality) is often denoted by the symbol \\, \\equiv \\, <\/span>. Using this notation, we write \\, 3 + 5 \\equiv 8 \\, <\/span>, and \\, (x + y) (x - y) \\equiv x^2 - y^2 \\, <\/span>. <\/p>\n

IMPORTANT<\/strong>: The second example above is only valid if the algebra is commutative<\/a>,
\nsince the expansion of the left-hand side gives (using the distributive property twice): \\, (x + y) (x - y) \\equiv x(x-y) + y(x-y) \\equiv x^2 - xy + yx - y^2 <\/span>,
\nwhich is identically equal to the right-hand side if-and-only-if \\, xy \\equiv yx <\/span>.<\/p>\n

2) Conditional<\/strong> (or equational<\/strong>) equality:<\/p>\n

Examples<\/strong>: The values of \\, x \\, <\/span> that satisfy the equation \\, 3x^2 - 5x + 2 = 0 \\, <\/span>,
\nand the values of\u00a0 \\, x \\, <\/span> and \\, y \\, <\/span> that satisfy the equation \\, 3x + 5y = 2 \\, <\/span>. <\/p>\n

3) Relational<\/strong> or equivalence <\/strong> equality:<\/p>\n

Example<\/strong>: \\, x = y \\, <\/span> if-and-only-if \\, x - y \\, <\/span> is divisible by \\, 7. <\/span><\/p>\n

Notation<\/strong>: This type of equality is often denoted by the symbol \\, \\cong \\, <\/span>.
\nUsing this notation, we can express our example as \\, x \\cong y \\, <\/span> if \\, x - y \\, <\/span> is divisible by \\, 7. <\/span><\/p>\n

4) Defining<\/strong> equality: <\/p>\n

Example<\/strong>: \\, C \\, <\/span> is defined to be equal to \\, A + B \\, <\/span>.<\/p>\n

Notation<\/strong>: \\, C \\stackrel {\\mathrm{def}}{=} A + B. <\/span> <\/p>\n

5) Assigned<\/strong> equality:<\/p>\n

Example<\/strong>: \\, R \\, <\/span> is assigned the value of \\, P + Q \\, <\/span>.<\/p>\n

Notation<\/strong>: Assigned equality is often denoted by the symbol \\, := \\, <\/span>
\nand using this notation we can express the example as \\, R := P + Q \\, <\/span>. <\/p>\n

\/\/\/\/\/\/\/ <\/p>\n","protected":false},"excerpt":{"rendered":"

This page is a sub-page of our page on Disambiguation. \/\/\/\/\/\/\/ Like the term ‘add’ (represented by the sign), the term ‘equal’ (represented by the sign) has many different meanings in mathematics. In fact, the sign can stand for at least five different types of equality: 1) Identical (or algebraic) equality: Examples: , and . … Continue reading Disambiguating equality<\/span> →<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"parent":4261,"menu_order":0,"comment_status":"open","ping_status":"open","template":"","meta":{"footnotes":""},"class_list":["post-995","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/kmr.placify.me\/wp\/wp-json\/wp\/v2\/pages\/995","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/kmr.placify.me\/wp\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/kmr.placify.me\/wp\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/kmr.placify.me\/wp\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/kmr.placify.me\/wp\/wp-json\/wp\/v2\/comments?post=995"}],"version-history":[{"count":0,"href":"https:\/\/kmr.placify.me\/wp\/wp-json\/wp\/v2\/pages\/995\/revisions"}],"up":[{"embeddable":true,"href":"https:\/\/kmr.placify.me\/wp\/wp-json\/wp\/v2\/pages\/4261"}],"wp:attachment":[{"href":"https:\/\/kmr.placify.me\/wp\/wp-json\/wp\/v2\/media?parent=995"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}