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denir."]},"beginTime":656,"endTime":719,"href":"/video/preview/13794142072813210977?parent-reqid=1766697747761668-7504711439666150863-balancer-l7leveler-kubr-yp-vla-68-BAL&text=Abstract+Theory+-+Topic&t=656&ask_summarization=1"},{"index":6,"title":"Etkisiz Eleman","list":{"type":"unordered","items":["Tam sayıların toplama işlemine göre etkisiz elemanı sıfırdır, çünkü herhangi bir tam sayı ile toplandığında o tam sayıyı etkilemez.","Reel sayıların çarpmaya göre etkisiz elemanı birdir, çünkü herhangi bir reel sayı ile çarpıldığında o reel sayıyı etkilemez.","Teorem 1.13'e göre bir cebirsel yapıda etkisiz eleman varsa, bu eleman tek türlü olarak belirlidir ve birbirinden farklı iki etkisiz eleman olamaz."]},"beginTime":724,"endTime":886,"href":"/video/preview/13794142072813210977?parent-reqid=1766697747761668-7504711439666150863-balancer-l7leveler-kubr-yp-vla-68-BAL&text=Abstract+Theory+-+Topic&t=724&ask_summarization=1"},{"index":7,"title":"Ters Eleman","list":{"type":"unordered","items":["Bir cebirsel yapıda etkisiz eleman (e) bulunduğunda, bir a elemanının tersi (a*) o elemana hem soldan hem sağdan çarpıldığında etkisiz elemanı veren elemandır.","Z kümesinde toplamaya göre bir a tam sayısının tersi -a'dır, çünkü a + (-a) = 0'dır.","Q kümesinde çarpmaya göre bir sıfırdan farklı alfa rasyonel sayısının tersi 1/alfa'dır, çünkü alfa × (1/alfa) = 1'dir."]},"beginTime":886,"endTime":1107,"href":"/video/preview/13794142072813210977?parent-reqid=1766697747761668-7504711439666150863-balancer-l7leveler-kubr-yp-vla-68-BAL&text=Abstract+Theory+-+Topic&t=886&ask_summarization=1"},{"index":8,"title":"Ters Elemanın Tekliği","list":{"type":"unordered","items":["Teorem 1.16'e göre birleşme özelliğine ve etkisiz elemana sahip bir cebirsel yapıda bir elemanın en fazla bir tane tersi vardır.","Bir elemanın tersi ya yoktur ya da tektir, birbirinden farklı iki tersi olamaz.","Z kümesinde çarpmaya göre 2'nin tersi yoktur, çünkü 2 × x = 1 koşulunu sağlayan bir tam sayı x bulunmaz."]},"beginTime":1107,"endTime":1271,"href":"/video/preview/13794142072813210977?parent-reqid=1766697747761668-7504711439666150863-balancer-l7leveler-kubr-yp-vla-68-BAL&text=Abstract+Theory+-+Topic&t=1107&ask_summarization=1"},{"index":9,"title":"Yarı Grup ve Grup","list":{"type":"unordered","items":["Birleşme özelliğine sahip bir ikili işlem tanımlanmış olan tek işlemli bir cebirsel yapıya yarı grup denir.","Yarı grup için iki özellik gerekli: işlem tanımlı olmalı ve birleşme özelliği sağlanmalı.","Grup için dört özellik gerekli: kapalılık, birleşme, sol etkisiz eleman ve her elemanın sol tersi olması."]},"beginTime":1271,"endTime":1467,"href":"/video/preview/13794142072813210977?parent-reqid=1766697747761668-7504711439666150863-balancer-l7leveler-kubr-yp-vla-68-BAL&text=Abstract+Theory+-+Topic&t=1271&ask_summarization=1"},{"index":10,"title":"Grup Kavramının Tanımı","list":{"type":"unordered","items":["Grup kavramının tanımı farklı şekillerde verilebilir; örneğin, üç aksiyon yerine sadece sol etkisiz eleman veya dörtte sadece sol ters yerine ters eleman (hem sol hem sağdan ters) istenebilir.","Eğer dört özellik sağlanırsa, e elemanı hem sol hem sağdan etkisiz eleman olacak ve a-yıldız elemanı ters eleman olacaktır.","Tam sayıların toplama işlemine göre oluşturduğu yapı, grup tanımını sağlar: kapalılık, birleşme, etkisiz eleman (0) ve ters eleman (a için -a) özellikleri mevcuttur."]},"beginTime":1476,"endTime":1653,"href":"/video/preview/13794142072813210977?parent-reqid=1766697747761668-7504711439666150863-balancer-l7leveler-kubr-yp-vla-68-BAL&text=Abstract+Theory+-+Topic&t=1476&ask_summarization=1"},{"index":11,"title":"Grup Örnekleri","list":{"type":"unordered","items":["Tam sayılar toplama işlemine göre bir grup oluştururken, rasyonel sayılar ve reel sayılar da benzer şekilde toplama işlemine göre grup oluştururlar.","Karmaşık sayılar kümesi (a+bi) toplama işlemine göre bir grup oluşturur; etkisiz elemanı 0+0i'dir ve her karmaşık sayının tersi vardır.","Her grup bir yarı gruptur, ancak her yarı grubun bir grup olması gerekmez; örneğin, tam sayılar çarpma işlemine göre yarı grup olmasına rağmen bir grup değildir."]},"beginTime":1653,"endTime":1856,"href":"/video/preview/13794142072813210977?parent-reqid=1766697747761668-7504711439666150863-balancer-l7leveler-kubr-yp-vla-68-BAL&text=Abstract+Theory+-+Topic&t=1653&ask_summarization=1"},{"index":12,"title":"Değişmeli Gruplar","list":{"type":"unordered","items":["Bir yarı grubun içindeki işlem değişmeli ise, o değişmeli yarı gruptur.","Bir grubun içindeki işlem değişmeli ise, o değişmeli grup veya Abel grubudur.","Verilen örnekler (tam sayılar, rasyonel sayılar, reel sayılar ve karmaşık sayılar toplama işlemine göre) hepsi değişmeli gruptur."]},"beginTime":1856,"endTime":1935,"href":"/video/preview/13794142072813210977?parent-reqid=1766697747761668-7504711439666150863-balancer-l7leveler-kubr-yp-vla-68-BAL&text=Abstract+Theory+-+Topic&t=1856&ask_summarization=1"},{"index":13,"title":"Grup Tanımındaki Özelliklerin İspatı","list":{"type":"unordered","items":["Grup olmak için kapalılık, birleşme, sol etkisiz ve sol ters eleman istenir, ancak bu özelliklerin kendiliğinden sağlanması gerekir.","Teorem 1.25'e göre, bir grubun tanımındaki a-yıldız elemanı, sadece a işlem a = e koşulunu sağlarken aynı zamanda a işlem a-yıldız = e koşulunu da sağlar.","İspat sonucunda, a-yıldız elemanı hem sol hem sağdan ters eleman olduğu, yani a'nın tersi olduğu görülür."]},"beginTime":1935,"endTime":2181,"href":"/video/preview/13794142072813210977?parent-reqid=1766697747761668-7504711439666150863-balancer-l7leveler-kubr-yp-vla-68-BAL&text=Abstract+Theory+-+Topic&t=1935&ask_summarization=1"},{"index":14,"title":"Grup Tanımı ve Etkisiz Eleman","list":{"type":"unordered","items":["Grup tanımında e elemanı soldan etkisiz kalır ve her a elemanı için a * e = a koşulunu sağlar.","Her grubun bir anlamda etkisiz elemanı vardır ve bu eleman hem soldan hem sağdan etkisizdir.","Teorem 1.25'e göre, her a elemanının sol tersi aynı zamanda sağ tersi de olur ve her grubun tek türlü belirli bir etkisiz elemanı vardır."]},"beginTime":2190,"endTime":2329,"href":"/video/preview/13794142072813210977?parent-reqid=1766697747761668-7504711439666150863-balancer-l7leveler-kubr-yp-vla-68-BAL&text=Abstract+Theory+-+Topic&t=2190&ask_summarization=1"},{"index":15,"title":"Ters Elemanların Varlığı","list":{"type":"unordered","items":["Her grubun daima tek türlü belirli bir etkisiz elemanı vardır ve her elemanın tek türlü belirli bir tersi vardır.","Grubun tanımı her elemanın bir sol tersi olduğunu söylerken, teorem 1.25 sol tersin aynı zamanda sağ ters olduğunu gösterir.","Birleşme özelliğine sahip ve etkisiz elemana sahip bir cebirsel yapıda ters elemanın tek türlü belirli olduğunu göstermişizdir."]},"beginTime":2329,"endTime":2385,"href":"/video/preview/13794142072813210977?parent-reqid=1766697747761668-7504711439666150863-balancer-l7leveler-kubr-yp-vla-68-BAL&text=Abstract+Theory+-+Topic&t=2329&ask_summarization=1"},{"index":16,"title":"Grup Tanımının Alternatif Formu","list":{"type":"unordered","items":["Teorem 1.28, grubu farklı bir şekilde tanımlar: kapalılık, birleşme ve her a, b çifti için a * x = b ve y * a = b denklemlerinin çözümlerinin var olması.","Gereklilik kısmında, g'nin bir grup olduğu varsayılarak üç koşulun sağlandığı gösterilir.","Tersine, bu üç özellik sağlanıyorsa g'nin bir grup olduğu gösterilir."]},"beginTime":2385,"endTime":2613,"href":"/video/preview/13794142072813210977?parent-reqid=1766697747761668-7504711439666150863-balancer-l7leveler-kubr-yp-vla-68-BAL&text=Abstract+Theory+-+Topic&t=2385&ask_summarization=1"},{"index":17,"title":"Etkisiz Elemanın Teklik İspatı","list":{"type":"unordered","items":["Etkisiz elemanın tekliğini ispatlamak için, herhangi bir a elemanı için e * a = a olduğunu göstermek gerekir.","Üçüncü aksiyomdan, her a ve b için a * x = b ve y * a = b denklemlerinin çözümleri olduğunu biliyoruz.","Bu şekilde, e elemanının bütün a elemanları için etkisiz olduğunu göstererek etkisiz elemanın tekliğini ispatlarız."]},"beginTime":2613,"endTime":2772,"href":"/video/preview/13794142072813210977?parent-reqid=1766697747761668-7504711439666150863-balancer-l7leveler-kubr-yp-vla-68-BAL&text=Abstract+Theory+-+Topic&t=2613&ask_summarization=1"},{"index":18,"title":"Grup Tanımlarının Farklı Şekilleri","list":{"type":"unordered","items":["Grup farklı şekillerde tanımlanabilir: kapalılık, birleşme, sol etkisiz ve sol ters eleman veya kapalılık, birleşme, etkisiz eleman ve ters eleman.","Teorem 1.28'e göre, grup kapalılık, birleşme ve her a, b elemanı için a * x = b ve y * a = b denklemlerinin çözümlerinin var olması koşullarını sağlar.","Grup ispatlarında işlem notasyonu kolaylaştırmak için çarpımsal notasyon kullanılır ve etkisiz eleman 1, ters eleman a⁻¹ şeklinde gösterilir."]},"beginTime":2772,"endTime":2891,"href":"/video/preview/13794142072813210977?parent-reqid=1766697747761668-7504711439666150863-balancer-l7leveler-kubr-yp-vla-68-BAL&text=Abstract+Theory+-+Topic&t=2772&ask_summarization=1"}],"linkTemplate":"/video/preview/13794142072813210977?parent-reqid=1766697747761668-7504711439666150863-balancer-l7leveler-kubr-yp-vla-68-BAL&text=Abstract+Theory+-+Topic&t=%%timestamp%%&ask_summarization=1"},"isAdultDoc":false,"relatedParams":{"text":"Video 1: Introduction to Group Theory - Abstract Mathematics - Algebra 1 (Undergraduate Course)","related_orig_text":"Abstract Theory - Topic","related_porno":false,"related_less_3m_off":true,"client":"d2d","no_cnt":1,"related_src":"serp","related":"{\"porno\":false,\"vfp\":1,\"orig_text\":\"Abstract Theory - 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