{"id":123,"date":"2024-08-23T14:12:49","date_gmt":"2024-08-23T14:12:49","guid":{"rendered":"https:\/\/kierunekmatura.pl\/blog\/?p=123"},"modified":"2025-08-01T20:24:23","modified_gmt":"2025-08-01T20:24:23","slug":"zbiory","status":"publish","type":"post","link":"https:\/\/kierunekmatura.pl\/blog\/zbiory\/","title":{"rendered":"Zbiory"},"content":{"rendered":"

1. Zbiory – wprowadzenie<\/p>

\r\n

\r\nZbi\u00f3r<\/span><\/h3>\r\n

Zbi\u00f3r to kolekcja dobrze okre\u015blonych obiekt\u00f3w, zwanych elementami zbioru. Elementy te mog\u0105 by\u0107 dowolnej natury: liczby, osoby, przedmioty itp.<\/p>\n\r\n<\/div>

\r\n

Przyk\u0142ad<\/span><\/h4>\r\n

Przyk\u0142adowe zbiory:
\n* A = {1, 2, 3} – zbi\u00f3r liczb naturalnych od 1 do 3
\n* B = {a, b, c, …, z} – zbi\u00f3r liter alfabetu \u0142aci\u0144skiego
\n* C = {x | x jest liczb\u0105 parzyst\u0105} – zbi\u00f3r liczb parzystych<\/p>\n\r\n<\/div>

2. Symbole i oznaczenia<\/p>

\r\n

\r\nSymbole zwi\u0105zane ze zbiorami<\/span><\/h3>\r\n

* \u2208 – nale\u017cy do (np. 2 \u2208 A, czyli 2 jest elementem zbioru A)
\n* \u2209 – nie nale\u017cy do (np. 5 \u2209 A, czyli 5 nie jest elementem zbioru A)
\n* \u2282 – zawiera si\u0119 w (np. {1, 2} \u2282 A, czyli ka\u017cdy element zbioru {1, 2} jest te\u017c elementem zbioru A)
\n* \u2284 – nie zawiera si\u0119 w
\n* \u2205 – zbi\u00f3r pusty (zbi\u00f3r niezawieraj\u0105cy \u017cadnych element\u00f3w)<\/p>\n\r\n<\/div>

3. Rodzaje zbior\u00f3w<\/p>

\r\n

\r\nRodzaje zbior\u00f3w<\/span><\/h3>\r\n

* Zbi\u00f3r sko\u0144czony – zbi\u00f3r posiadaj\u0105cy sko\u0144czon\u0105 liczb\u0119 element\u00f3w (np. zbi\u00f3r dni tygodnia)
\n* Zbi\u00f3r niesko\u0144czony – zbi\u00f3r posiadaj\u0105cy niesko\u0144czenie wiele element\u00f3w (np. zbi\u00f3r liczb naturalnych)
\n* Zbi\u00f3r pusty – zbi\u00f3r niezawieraj\u0105cy \u017cadnych element\u00f3w<\/p>\n\r\n<\/div>

4. Relacje mi\u0119dzy zbiorami<\/p>

\r\n

\r\nRelacje mi\u0119dzy zbiorami<\/span><\/h3>\r\n

* R\u00f3wno\u015b\u0107 zbior\u00f3w (A = B) – zachodzi, gdy zbiory A i B maj\u0105 dok\u0142adnie te same elementy.
\n* Roz\u0142\u0105czno\u015b\u0107 zbior\u00f3w – zbiory A i B s\u0105 roz\u0142\u0105czne, gdy nie maj\u0105 \u017cadnych wsp\u00f3lnych element\u00f3w (A \u2229 B = \u2205).
\n* Podzbi\u00f3r – zbi\u00f3r A jest podzbiorem zbioru B (A \u2282 B), je\u017celi ka\u017cdy element zbioru A jest tak\u017ce elementem zbioru B.<\/p>\n\r\n<\/div>

\r\n

Przyk\u0142ad<\/span><\/h4>\r\n

Niech A = {1, 2, 3}, B = {2, 4}, C = {1, 2, 3, 4}.
\n* A \u2260 B (zbiory A i B nie s\u0105 r\u00f3wne)
\n* A i B s\u0105 roz\u0142\u0105czne (nie maj\u0105 wsp\u00f3lnych element\u00f3w)
\n* A \u2282 C (A jest podzbiorem C)<\/p>\n\r\n<\/div>

5. Sposoby opisu zbior\u00f3w<\/p>

\r\n

\r\nSposoby opisu zbior\u00f3w<\/span><\/h3>\r\n

Zbi\u00f3r mo\u017cna opisa\u0107 na kilka sposob\u00f3w:
\n* Wyliczenie element\u00f3w: A = {1, 2, 3}
\n* Opis s\u0142owny: A to zbi\u00f3r liczb naturalnych od 1 do 3
\n* Notacja matematyczna: A = {x \u2208 N : 1 \u2264 x \u2264 3} (czytamy: A to zbi\u00f3r takich x nale\u017c\u0105cych do zbioru liczb naturalnych, \u017ce x jest wi\u0119ksze lub r\u00f3wne 1 i mniejsze lub r\u00f3wne 3)<\/p>\n\r\n<\/div>

\r\n

Przyk\u0142ad<\/span><\/h4>\r\n

* Opis s\u0142owny: A to zbi\u00f3r takich liczb naturalnych, kt\u00f3rych kwadrat jest mniejszy lub r\u00f3wny 9
\n* Wyliczenie element\u00f3w: A = {0, 1, 2, 3}
\n* Notacja matematyczna: A = {a \u2208 N : a\u00b2 \u2264 9}<\/p>\n\r\n<\/div>","protected":false},"excerpt":{"rendered":"

1. Zbiory – wprowadzenie 2. Symbole i oznaczenia 3. Rodzaje zbior\u00f3w 4. Relacje mi\u0119dzy zbiorami 5. Sposoby opisu zbior\u00f3w<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[106],"tags":[],"class_list":["post-123","post","type-post","status-publish","format-standard","hentry","category-matematyka"],"_links":{"self":[{"href":"https:\/\/kierunekmatura.pl\/blog\/wp-json\/wp\/v2\/posts\/123","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/kierunekmatura.pl\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/kierunekmatura.pl\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/kierunekmatura.pl\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/kierunekmatura.pl\/blog\/wp-json\/wp\/v2\/comments?post=123"}],"version-history":[{"count":2,"href":"https:\/\/kierunekmatura.pl\/blog\/wp-json\/wp\/v2\/posts\/123\/revisions"}],"predecessor-version":[{"id":525,"href":"https:\/\/kierunekmatura.pl\/blog\/wp-json\/wp\/v2\/posts\/123\/revisions\/525"}],"wp:attachment":[{"href":"https:\/\/kierunekmatura.pl\/blog\/wp-json\/wp\/v2\/media?parent=123"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/kierunekmatura.pl\/blog\/wp-json\/wp\/v2\/categories?post=123"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/kierunekmatura.pl\/blog\/wp-json\/wp\/v2\/tags?post=123"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}