חוק הפילוג

(המסומנת בבמתמטיקה (אלגברה), חוק הפילוג הוא חוק המקשר בין שתי פעולות בינאריות. דוגמה ידועה היא חוק הפילוג של פעולת הכפל (המסומנת ב $\cdot$ ) מעל פעולת החיבור (המסומנת ב $+$ ), או כי הכפל דיסטריבוטיבי מעל החיבור: $a\cdot (b+c)=(a\cdot b)+(a\cdot c)$ .

ניתן לראות דוגמה לתכונה זו במשוואה הבאה:

4\cdot (2+3)=(4\cdot 2)+(4\cdot 3)

כאן באגף שמאל של המשוואה, 4 מוכפל בסכום של שני מספרים, 3 ו-2. לעומת זאת, בצד ימין 4 מוכפל בכל אחד מהאיברים בנפרד, ואת התוצאות מחברים. מכיוון שבשתי הדרכים מתקבלת אותה התוצאה, ולא משנה איזה מספרים יוצבו במשוואה, אומרים במקרה זה כי חוק הפילוג תקף, או כי הכפל דיסטריבוטיבי מעל החיבור.

נשים לב שהדבר אינו תמיד נכון לכל שתי פעולות בינאריות, למשל:

4+(2\cdot 3)\neq (4+2)\cdot (4+3)

כאן באגף שמאל התוצאה המתקבלת היא 10, ואילו באגף ימין מתקבל 42. על כן החיבור אינו דיסטריבוטיבי מעל הכפל.

הגדרה פורמלית

תהא $\!\,S$ קבוצה ויהיו $\!\,*,+$ שתי פעולות בינאריות המוגדרות עליה. נאמר כי:

$\!\,*$ הוא דיסטריבוטיבי משמאל מעל $\!\,+$ אם לכל $\!\,x,y,z\in S$ מתקיים $\!\,x*(y+z)=(x*y)+(x*z)$ .
$\!\,*$ הוא דיסטריבוטיבי מימין מעל $\!\,+$ אם לכל $\!\,x,y,z\in S$ מתקיים $\!\,(y+z)*x=(y*x)+(z*x)$ .
$\!\,*$ הוא דיסטריבוטיבי מעל $\!\,+$ אם הוא דיסטריבוטיבי מימין ומשמאל מעליו.

אם $\!\,*$ היא פעולה קומוטטיבית הרי ששלושת התנאים הללו שקולים.

דוגמאות

הכפל מתפלג מעל החיבור, וכן מעל החיסור. החילוק מתפלג מעל שתי הפעולות הנ"ל מימין.
החזקה מתפלגת משמאל מעל הכפל.
האיחוד והחיתוך מתפלגים זה מעל זה.
באלגברה בוליאנית, פעולות ה"וגם" וה"או" מתפלגות זו מעל זו.

קישורים חיצוניים

חוק הפילוג, באתר MathWorld (באנגלית)

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