prove it - the total number of subsets of a finite set containing n elements is 2^n

Asked by nk.pandit02 | 29th May, 2016, 01:42: PM

Expert Answer:

A space s u b s e t space i s space f o r m e d space b y space t a k i n g space e i t h e r space 1 space 0 r space 2 space o r space 3 space a n d space s o space o n space n u m b e r space o f space
e l e m e n t s space f r o m space a space g i u v e n space s e t. space ϕ space i s space a l s o space a space s u b s e t space o f space e v e r y space s e t space w h i c h space i s space
o b t a i n e d space b y space t a k i n g space n o n e space o f space t h e space e l e m e n t s. space
I n space t h e space l a n g u a g e space o f space c o u n t i n g comma space a space s u b s e t space i s space f o r m e d space b y space s e l e c t i n g space e i t h e r space 0 space o r space 1 space
o r space 2 space.... o r space a l l space apostrophe n apostrophe space e l e m e n t s space f r o m space a space s e t space o f space apostrophe n apostrophe space e l e m e n t s.
T h e space n u m b e r space o f space w a y s space o f space s e l e c t i n g space n o n e space i s space C presuperscript n subscript o
S i m i l a l r y comma space t h e space n u m b e r space o f space w a y s space o f space s e l e c t i n g space 1 space i s space C presuperscript n subscript 1
T h e space n u m b e r space o f space w a y s space o f space s e l e c t i n g space 2 space i s space C presuperscript n subscript 2
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T h e space n u m b e r space o f space w a y s space o f space s e l e c t i n g space n space i s space C presuperscript n subscript n
H e n c e comma space t h e space t o t a l space n u m b e r space o f space s u b s e t s space p o s s i b l e space i s
space space C presuperscript n subscript o plus C presuperscript n subscript 1 plus space C presuperscript n subscript 2 plus... plus C presuperscript n subscript n
B Y space p r o p e r t y space o f space s u m space o f space b i n o m i a l space c o e f f i c i e n t s
C presuperscript n subscript o plus C presuperscript n subscript 1 plus space C presuperscript n subscript 2 plus... plus C presuperscript n subscript n equals 2 to the power of n
H e n c e comma space t h e space t o t a l space n u m b e r space o f space s u b s e t s space o f space a space g i v e n space s e t space o f space apostrophe n apostrophe space e l e m e n t s space i s space 2 to the power of n. space left parenthesis p r o v e d right parenthesis

Answered by satyajit samal | 29th May, 2016, 05:58: PM