K\u00fcme, nesnelerin iyi tan\u0131mlanm\u0131\u015f listesidir.K\u00fcmeler genellikle A, B, C gibi b\u00fcy\u00fck harflerle g\u00f6sterilir. <\/p>\n
K\u00fcmeyi olu\u015fturan \u00f6gelere, k\u00fcmenin eleman\u0131 denir. a eleman\u0131 A k\u00fcmesine ait ise,
\n a \u00ce A bi\u00e7iminde yaz\u0131l\u0131r. \u201ca, A k\u00fcmesinin eleman\u0131d\u0131r.\u201d diye okunur. b eleman\u0131 A k\u00fcmesine ait de\u011filse, b \u00cf A bi\u00e7iminde yaz\u0131l\u0131r. \u201cb, A k\u00fcmesinin eleman\u0131 de\u011fildir.\u201d diye okunur.<\/p>\n
K\u00fcmede, ayn\u0131 eleman bir kez yaz\u0131l\u0131r.<\/p>\n
Elemanlar\u0131n yerlerinin de\u011fi\u015ftirilmesi k\u00fcmeyi de\u011fi\u015ftirmez.<\/p>\n
A k\u00fcmesinin eleman say\u0131s\u0131 s(A) ya da n(A) ile g\u00f6sterilir.<\/p>\n
B. K\u00dcMELER\u0130N G\u00d6STER\u0130L\u0130\u015e\u0130<\/p>\n
K\u00fcmenin elemanlar\u0131 a\u015fa\u011f\u0131daki 3 yolla g\u00f6sterilebilir.<\/p>\n
1. Liste Y\u00f6ntemi<\/p>\n
K\u00fcmenin elemanlar\u0131 { } sembol\u00fc i\u00e7ine, her bir eleman\u0131n aras\u0131na virg\u00fcl konularak yaz\u0131l\u0131r.<\/p>\n
A = {a, b, {a, b, c}} \u015e s(A) = 3 t\u00fcr.<\/p>\n
2. Ortak \u00d6zellik Y\u00f6ntemi<\/p>\n
K\u00fcmenin elemanlar\u0131, daha somut ya da daha kolay alg\u0131lan\u0131r bi\u00e7imde gerekti\u011finde s\u00f6zel, gerekti\u011finde matematiksel bir ifade olarak ortaya koyma bi\u00e7imidir.<\/p>\n
A = {x : (x in \u00f6zelli\u011fi)}<\/p>\n
Burada \u201cx :\u201d ifadesi \u201c\u00f6yle x lerden olu\u015fur ki\u201d diye okunur.<\/p>\n
Bu ifade \u201cx |\u201d bi\u00e7iminde de yaz\u0131labilir.<\/p>\n
3. Venn \u015eemas\u0131 Y\u00f6ntemi<\/p>\n
K\u00fcme, kapal\u0131 bir e\u011fri i\u00e7inde her eleman bir nokta ile<\/p>\n
g\u00f6sterilip noktan\u0131n yan\u0131na eleman\u0131n ad\u0131 yaz\u0131larak<\/p>\n
g\u00f6sterilir.<\/p>\n
Bu g\u00f6sterime Venn \u015eemas\u0131 ile g\u00f6sterim denir.<\/p>\n
C. E\u015e\u0130T K\u00dcME, DENK K\u00dcME<\/p>\n
Ayn\u0131 elemanlardan olu\u015fan k\u00fcmelere e\u015fit k\u00fcmeler denir. Eleman say\u0131lar\u0131 e\u015fit olan k\u00fcmelere denk k\u00fcmeler denir.<\/p>\n
A k\u00fcmesi B k\u00fcmesine e\u015fit ise A = B,<\/p>\n
C k\u00fcmesi D k\u00fcmesine denk ise C \u00ba D<\/p>\n
bi\u00e7iminde g\u00f6sterilir.<\/p>\n
E\u015fit olan k\u00fcmeler ay\u0131n zamanda denktir. Fakat denk k\u00fcmeler e\u015fit olmayabilir.<\/p>\n
D. BO\u015e K\u00dcME<\/p>\n
Hi\u00e7 bir eleman\u0131 olmayan k\u00fcmeye bo\u015f k\u00fcme denir.<\/p>\n
Bo\u015f k\u00fcme { } ya da \u00c6 sembolleri ile g\u00f6sterilir.<\/p>\n
E\u015fit olan k\u00fcmeler ay\u0131n zamanda denktir. Fakat denk k\u00fcmeler e\u015fit olmayabilir.<\/p>\n
{.} ve {0} k\u00fcmeleri bo\u015f k\u00fcme olmay\u0131p birer elemana sahip iki denk k\u00fcmedir.<\/p>\n
{\u00c6} ve {0} k\u00fcmeleri bo\u015f k\u00fcme olmay\u0131p birer elemana sahip iki denk k\u00fcmedir.<\/p>\n
E. ALT K\u00dcME – \u00d6ZALT K\u00dcME<\/p>\n
1. Alt K\u00fcme<\/p>\n
A k\u00fcmesinin her eleman\u0131, B k\u00fcmesinin de eleman\u0131 ise A ya B nin alt k\u00fcmesi denir.<\/p>\n
A k\u00fcmesi B k\u00fcmesinin alt k\u00fcmesi ise A \u00cc B bi\u00e7iminde g\u00f6sterilir.<\/p>\n
A k\u00fcmesi B k\u00fcmesinin alt k\u00fcmesi ise B k\u00fcmesi A k\u00fcmesini kaps\u0131yor denir. B \u00c9 A bi\u00e7iminde g\u00f6sterilir.<\/p>\n
C k\u00fcmesi D k\u00fcmesinin alt k\u00fcmesi de\u011filse C \u00cb D bi\u00e7iminde g\u00f6sterilir.<\/p>\n
2. \u00d6zalt K\u00fcme<\/p>\n
Bir k\u00fcmenin, kendisinden farkl\u0131 b\u00fct\u00fcn alt k\u00fcmelerine o k\u00fcmenin \u00f6zalt k\u00fcmeleri denir.<\/p>\n
3. Alt K\u00fcmenin \u00d6zellikleri<\/p>\n
i) Her k\u00fcme kendisinin alt k\u00fcmesidir.<\/p>\n
A \u00cc A<\/p>\n
ii) Bo\u015f k\u00fcme her k\u00fcmenin alt k\u00fcmesidir.<\/p>\n
\u00c6 \u00cc A<\/p>\n
iii) (A \u00cc B ve B \u00cc A) \u00db A = B dir.<\/p>\n
\u0131v) (A \u00cc B ve B \u00cc C) \u015e A \u00cc C dir.<\/p>\n
v) n elemanl\u0131 bir k\u00fcmenin alt k\u00fcmelerinin say\u0131s\u0131 2n ve \u00f6zalt k\u00fcmelerinin say\u0131s\u0131 2n \u2013 1 dir.<\/p>\n
v\u0131) n elemanl\u0131 bir k\u00fcmenin r tane (n \u00b3 r) elemanl\u0131 alt k\u00fcmelerinin say\u0131s\u0131<\/p>\n
F. K\u00dcMELERLE YAPILAN \u0130\u015eLEMLER<\/p>\n
1. K\u00fcmelerin Birle\u015fimi<\/p>\n
A n\u0131n elemanlar\u0131ndan veya B nin elemanlar\u0131ndan olu\u015fan k\u00fcmeye bu iki k\u00fcmenin birle\u015fim k\u00fcmesi denir ve A \u00c8 B bi\u00e7iminde g\u00f6sterilir.<\/p>\n
A \u00c8 B = {x : x \u00ce A veya x \u00ce B} dir.<\/p>\n
2. Birle\u015fim I\u015fleminin \u00d6zellikleri<\/p>\n
i) A \u00c8 \u00c6 = A<\/p>\n
ii) A \u00c8 A = A<\/p>\n
iii) A \u00c8 B = B \u00c8 A<\/p>\n
\u0131v) A \u00c8 (B \u00c8 C) = (A \u00c8 B) \u00c8 C<\/p>\n
v) A \u00cc B ise, A \u00c8 B = B<\/p>\n
v\u0131) A \u00c8 B = \u00c6 ise, (A = \u00c6 ve B = \u00c6) dir.<\/p>\n
3. K\u00fcmelerin Kesi\u015fimi<\/p>\n
A ve B k\u00fcmesinin ortak elemanlar\u0131ndan olu\u015fan<\/p>\n
k\u00fcmeye A ile B nin kesi\u015fim k\u00fcmesi denir ve A \u00c7 B<\/p>\n
bi\u00e7iminde g\u00f6sterilir.<\/p>\n
A \u00c7 B = {x : x \u00ce A ve x \u00ce B} dir.<\/p>\n
4. Kesi\u015fim I\u015fleminin \u00d6zellikleri<\/p>\n
i) A \u00c7 \u00c6 = \u00c6<\/p>\n
ii) A \u00c7 A = A<\/p>\n
iii) A \u00c7 B = B \u00c7 A<\/p>\n
\u0131v) (A \u00c7 B) \u00c7 C = A \u00c7 (B \u00c7 C)<\/p>\n
v) A \u00c7 (B \u00c8 C) = (A \u00c7 B) \u00c8 (A \u00c7 C)<\/p>\n
v\u0131) A \u00c8 (B \u00c7 C) = (A \u00c8 B) \u00c7 (A \u00c8 C)<\/p>\n
G. EVRENSEL K\u00dcME<\/p>\n
\u00dczerinde i\u015flem yap\u0131lan, b\u00fct\u00fcn k\u00fcmeleri kapsayan k\u00fcmeye, evrensel k\u00fcme denir. Evrensel k\u00fcme genellikle E ile g\u00f6sterilir.<\/p>\n
H. B\u0130R K\u00dcMEN\u0130N T\u00dcMLEYEN\u0130<\/p>\n
Evrensel k\u00fcmenin eleman\u0131 olup, A k\u00fcmesinin eleman\u0131 olmayan elemanlardan olu\u015fan k\u00fcmeye A n\u0131n t\u00fcmleyeni denir ve A ya da A’ ile g\u00f6sterilir.<\/p>\n
A = {x : x \u00ce E ve x \u00cf A, A \u00cc E} dir.<\/p>\n
T\u00fcmleyenin \u00d6zellikleri<\/p>\n
i) E = \u00c6<\/p>\n
ii) \u00c6 = E<\/p>\n
iii) () = A<\/p>\n
iv) A \u00c8 A = E ve A \u00c7 A = \u00c6 dir.<\/p>\n
v) A \u00c8 B = A \u00c7 B<\/p>\n
v\u0131) A \u00c7 B = A \u00c8 B<\/p>\n
v\u0131\u0131) E \u00c8 A = E ve E \u00c7 A = A dir.<\/p>\n
v\u0131\u0131\u0131) A \u00cc B ise, B \u00cc A dir.<\/p>\n
I. KUVVET K\u00dcMESI<\/p>\n
Bir k\u00fcmenin b\u00fct\u00fcn alt k\u00fcmelerin k\u00fcmesine kuvvet k\u00fcmesi denir. Kuvvet k\u00fcmesi P(A) ile g\u00f6sterilir.<\/p>\n
s(A) = n ise, s(P(A)) = 2n dir.<\/p>\n
J. \u0130K\u0130 K\u00dcMEN\u0130N FARKI<\/p>\n
A k\u00fcmesinde olup, B k\u00fcmesinde olmayan elemanlar\u0131n k\u00fcmesine A fark B k\u00fcmesi denir. A fark B k\u00fcmesi A \u2013 B ya da A B bi\u00e7iminde g\u00f6sterilir.<\/p>\n
A \u2013 B = {x : x \u00ce A ve x \u00cf B} dir.<\/p>\n
Farkla Ilgili \u00d6zellikler<\/p>\n
A, B, C k\u00fcmeleri E evrensel k\u00fcmesinin alt k\u00fcmeleri olmak \u00fczere,<\/p>\n
i) E \u2013 A = A<\/p>\n
ii) A \u2013 B = A \u00c7 B<\/p>\n
iii) A \u2013 B = A \u00c8 B dir.<\/p>\n
\u0131v) (A \u2013 B) \u00c8 (B \u2013 A) = A D B (Simetrik Fark)<\/p>\n
K. ELEMAN SAYISI<\/p>\n
A, B, C herhangi birer k\u00fcme olmak \u00fczere,<\/p>\n
i) s(A \u00c8 B) = s(A) + s(B) \u2013 s(A \u00c7 B)<\/p>\n
ii) s(A \u00c8 B \u00c8 C) = s(A) + s(B) + s(C) \u2013 s(A \u00c7 B) \u2013 s(A \u00c7 C)<\/p>\n
\u2013 s(B \u00c7 C) + s(A \u00c7 B \u00c7 C)<\/p>\n
iii) s(A \u00c8 B) = s(A \u2013 B) + s(A \u00c7 B) + s(B \u2013 A)<\/p>\n
\u0131v) a + b + c + d tane \u00f6\u011frencinin bulundu\u011fu bir s\u0131n\u0131fta voleybol oynayan \u00f6\u011frencilerin say\u0131s\u0131 s(V) = b + c, tenis oynayan \u00f6\u011frencilerin say\u0131s\u0131 s(T) = a + b, voleybol ve tenis oynayan \u00f6\u011frencilerin say\u0131s\u0131 s(T \u00c7 V) = b olsun.<\/p>\n
Tenis veya voleybol oynayanlar\u0131n say\u0131s\u0131:<\/p>\n
s(T \u00c8 V) = a + b + c<\/p>\n
Tenis ya da voleybol oynayanlar\u0131n say\u0131s\u0131:<\/p>\n
s(T \u2013 V) + s(V \u2013 T) = a + c<\/p>\n
Sadece tenis oynayanlar\u0131n say\u0131s\u0131:<\/p>\n
s(T \u2013 V) = a<\/p>\n
Tenis oynamayanlar\u0131n say\u0131s\u0131:<\/p>\n
s(T) = c + d<\/p>\n
Bu iki oyundan en az birini oynayanlar\u0131n say\u0131s\u0131:<\/p>\n
s(T \u00c8 V) = a + b + c<\/p>\n
Bu iki oyundan en \u00e7ok birini oynayanlar\u0131n say\u0131s\u0131:<\/p>\n
s(A \u00c7 B) = s(A \u00c8 B) + s(T \u2013 V) + s(V \u2013 T) = d + a + c<\/p>\n
Bu iki oyundan hi\u00e7 birini oynamayanlar\u0131n say\u0131s\u0131:<\/p>\n
s(A \u00c8 B) = d<\/p>\n","protected":false},"excerpt":{"rendered":"
K\u00fcme, nesnelerin iyi tan\u0131mlanm\u0131\u015f listesidir.K\u00fcmeler genellikle A, B, C gibi b\u00fcy\u00fck harflerle g\u00f6sterilir. K\u00fcmeyi olu\u015fturan \u00f6gelere, k\u00fcmenin eleman\u0131 denir. a eleman\u0131 A k\u00fcmesine ait ise, a \u00ce A bi\u00e7iminde yaz\u0131l\u0131r. \u201ca, A k\u00fcmesinin eleman\u0131d\u0131r.\u201d diye okunur. b eleman\u0131 A k\u00fcmesine ait de\u011filse, b \u00cf A bi\u00e7iminde yaz\u0131l\u0131r. \u201cb, A k\u00fcmesinin eleman\u0131 de\u011fildir.\u201d diye okunur. K\u00fcmede, …<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1404,1403],"tags":[3543,3540,3541,3542,3546,3537,3545,3547,3538,3544,3539],"class_list":["post-1209","post","type-post","status-publish","format-standard","hentry","category-matematik-odevleri","category-odevler","tag-alt-kume","tag-bos-kume","tag-denk-kume","tag-esit-kume","tag-evrensel-kume","tag-kumeler","tag-kumelerin-birlesimi","tag-kuvvet-kumesi","tag-liste-yontemi","tag-ozalt-kume","tag-venn-semasi-yontemi"],"_links":{"self":[{"href":"https:\/\/www.islamidavet.com\/kutuphane\/wp-json\/wp\/v2\/posts\/1209","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.islamidavet.com\/kutuphane\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.islamidavet.com\/kutuphane\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.islamidavet.com\/kutuphane\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.islamidavet.com\/kutuphane\/wp-json\/wp\/v2\/comments?post=1209"}],"version-history":[{"count":0,"href":"https:\/\/www.islamidavet.com\/kutuphane\/wp-json\/wp\/v2\/posts\/1209\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.islamidavet.com\/kutuphane\/wp-json\/wp\/v2\/media?parent=1209"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.islamidavet.com\/kutuphane\/wp-json\/wp\/v2\/categories?post=1209"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.islamidavet.com\/kutuphane\/wp-json\/wp\/v2\/tags?post=1209"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}