Binary Operations, Groups, and Examples in Mathematics
Explore the concept of groups in mathematics, focusing on binary operations, associativity, identity elements, and invertible elements. Learn through examples and exercises to deepen your understanding of group theory.
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Presentation Transcript
Group Let G be non empty set and * be binary operatio on G ,THEN (G,*) iS group if satisfy the following condition
Group 1- the binary operation * is associative 2-there is an element e in G S.t e*x=x*e=x for all x in G 3- for each a in G there is an element a^-1 in Gs.t a*a^-1=a^-1*a=e
Examples 1-(Z,+) is group 2- (Z+,+) is not group sincr for each a in Z+ there is not a^-1 ;a+a^-1=0 3-(Q,x)is not group since 0 in Q but 1\0 is not in Q
Remark 1- let (G,*)be gp with identity ,a in G then 1-(a)^n=a*a*a* .*a if a in G&(G,*)IS gp,n in N 2-(a)^0=e 3-(a)^-k=(a^-1)^k;k in Z
Exersies 1- prove that (Q\{0},*) is group if a*b=(ab)/2 2- is (Z+,+)have identity element 3- let (R,*) is a m.s s.t a*b=a+b-1 Find identity element 4-let (G,*) m.s s.t a*b=a+b 2 ;a,b in Q FIND the invertable element
