12/16/2018 Group Theory Group Theory Properties of Real Topic No. 1 Numbers 1 2 Properties of Real Numbers Properties of Real Numbers Number Systems 0.131313…=0.13+ ℕ ={ 1, 2, 3, … } 0.0013+0.000013+… ℤ={…, -2, -1, 0, 1, 2, … } =13/100+13/10000+ ℚ={p/q | p, q ∊ ℤ and q≠0} 13/1000000+… ℚˊ= Set of Irrational =(13/100)(1+1/100+ Numbers 1/10000+…) ℝ=ℚ ∪ ℚˊ =(13/100)(100/99) =13/99 3 4 Properties of Real Numbers Properties of Real Numbers . e=2.718281828459045… ∊ ℚˊ . ∀ a, b, c ∊ ℝ, (ab)c=a(bc) . √2=1.414213562373095… ∊ ℚˊ . For instance, ((-2/3)4)√2=(-8/3) √2 =(-2/3)(4 √2) . √5=2.23606797749978… ∊ ℚˊ . For every a ∊ ℝ and 0 ∊ ℝ, a+0=a=0+a . ∀ a, b ∊ ℝ, a.b ∊ ℝ . For every a ∊ ℝ and 1 ∊ ℝ, a.1=a=1.a . ∀ a, b ∊ ℝ, a+b ∊ ℝ . For every a ∊ ℝ there exists -a ∊ ℝ such that . ∀ a, b, c ∊ ℝ, (a+b)+c=a+(b+c) a+(-a)=0=(-a)+a . For example, (1/4+3)+ √7=(13+4 √7)/4=1/4+(3+ √7) . For every a ∊ ℝ\{0} there exists 1/a ∊ ℝ\{0} such that a(1/a)=1=(1/a)a . ∀ a, b ∊ ℝ, a+b=b+a . ∀ a, b ∊ ℝ, a.b=b.a 5 6 1 12/16/2018 Group Theory Group Theory Properties of Complex Topic No. 2 Numbers 7 8 Properties of Complex Numbers Properties of Complex Numbers . ℂ={a+bi | a, b ∊ ℝ} . ∀ a+bi, c+di, e+fi ∊ ℂ, [(a+bi).(c+di)].(e+fi) . ∀ a+bi, c+di ∊ ℂ, (a+bi)+(c+di)=(a+c)+(b+d)i ∊ ℂ =[(ac-bd)+(bc+ad)i].(e+fi) . ∀ a+bi, c+di ∊ ℂ, (a+bi).(c+di)=(ac-bd)+(ad+bc)i ∊ ℂ =[(ac-bd)e-(bc+ad)f]+[(bc+ad)e+(ac-bd)f]i . ∀ a+bi, c+di, e+fi ∊ ℂ, [(a+bi)+(c+di)]+(e+fi)= =[a(ce-df)-b(de+cf)]+[a(de+cf)]+b(ce-df)]i [(a+c)+(b+d)i]+(e+fi)=[(a+c)+e]+[(b+d)+f]i =(a+bi).[(ce-df)+(de+cf)i]=(a+bi).[(c+di).(e+fi)] =[a+(c+e)]+[b+(d+f)]i=(a+bi)+[(c+e)+(d+f)i]= . For every a+bi ∊ ℂ and 0=0+0i ∊ ℂ, (a+bi)+0= (a+bi)+[(c+di)+(e+fi)] (a+bi)+(0+0i)=(a+0)+(b+0)i=a+bi=0+(a+bi) . For every a+bi ∊ ℂ and 1=1+0i ∊ ℂ, (a+bi).1= (a+bi).(1+0i)=(a.1-0b)+(b.1+0.a)i=a+bi=1.(a+bi) 9 10 Properties of Complex Numbers Properties of Complex Numbers . For every a+bi ∊ ℂ there exists -a-bi ∊ ℂ such that . ∀ a+bi, c+di ∊ ℂ, (a+bi)+(-a-bi)=(a+(-a))+(b+(-b))i=0+0i=0=(-a-bi)+(a+bi) (a+bi)+(c+di)=(a+c)+(b+d)i . For every a+bi ∊ ℂ\{0} there exists =(c+a)+(d+b)i=(c+di)+(a+bi) 1/(a+bi)=a/(a2+b2)-(b/(a2+b2))i ∊ ℂ\{0} . ∀ a+bi, c+di ∊ ℂ, such that (a+bi).(a/(a2+b2)-(b/(a2+b2))i ) (a+bi).(c+di) = (a2+b2)/(a2+b2)+((ab-ab)/(a2+b2))i=1+0i=1 =(ac-bd)+ (ad+bc)i =(a/(a2+b2)-(b/(a2+b2))i )(a+bi) =(ca-db)+(cb+da)i =(c+di).(a+bi) 11 12 2 12/16/2018 Group Theory Group Theory Binary Operations Topic No. 3 13 14 Binary Operations Binary Operations Definition . Usual addition ‘+’ is a ∗ binary operation on the A binary operation on a + + set S is a function sets ℝ, ℂ, ℚ, ℤ, ℝ , ℚ , + mapping S x S into S. ℤ For each (a, b) ∈ S x S, we . Usual multiplication ‘.’ is a binary operation on will denote the element + ∗((a, b)) of S by a∗b. the sets ℝ, ℂ, ℚ, ℤ, ℝ , ℚ+, ℤ+ . Usual multiplication ‘.’ is a binary operation on the sets ℝ\{0}, ℂ\{0}, ℚ\{0}, ℤ\{0} 15 16 Binary Operations Binary Operations Let M(ℝ) be the set of all Usual addition ‘+’ is not a matrices with real entries. binary operation on the The usual matrix addition sets ℝ\{0}, ℂ\{0}, ℚ\{0}, is not a binary operation ℤ\{0} since on this set since A+B is 2+(-2)=0 ∉ ℤ\{0}⊂ℚ\{0} not defined for an ⊂ ℝ\{0} ⊂ ℂ\{0}. ordered pair (A, B) of matrices having different numbers of rows or of columns. 17 18 3 12/16/2018 Binary Operations Binary Operations Definition Usual addition ‘+’ on the Let ∗ be a binary set ℝ of real numbers operation on S and let H does not induce a binary be a subset of S. operation on the set ℝ\{0} of nonzero real The subset H is closed numbers because under ∗ if for all a, b ∈ H 2∈ℝ\{0} and -2∈ℝ\{0}, we also have a∗b ∈ H. but 2+(-2)=0 ∉ ℝ\{0}. In this case, the binary operation on H given by Thus ℝ\{0} is not restricting ∗ to H is the closed under +. induced operation of ∗ on H. 19 20 Binary Operations Group Theory Usual multiplication ‘.’ on the sets ℝ and ℚ induces a binary operation on the sets ℝ\{0}, ℝ+ and ℚ\{0}, Binary Operations ℚ+, respectively. 21 22 Binary Operations Binary Operations . Let S be a set and . Let be a set and a,. b S a,. b S . A binary operation on is a rule which assigns to any ordered pair (,) ab an element a b S. 23 24 4 12/16/2018 Binary Operations Binary Operations Examples Examples . For S ,,,,, . For S ,,,,, a b a b . For S ,,,,, a b ab . 25 26 Binary Operations Binary Operations Examples Examples . For S ,,,,, . For S ,,,,, . For S ,,,,, . For S ,,,,, . For . For S ,,,, S ,,,, a b a b . For S ,,,, a bmin( a , b ) 27 28 Binary Operations Binary Operations Examples Examples . For S 1,2,3 . For a b b a b b . For example 1 2 2, 1 1 1, 2 3 3. 29 30 5 12/16/2018 Binary Operations Binary Operations Examples Examples . For is not everywhere . For S , is not everywhere a b a/ b defined since no rational number is assigned by defined since no rational number is assigned by this rule to the pair this rule to the pair (3,0). For S , is not a binary operation on since is not closed under . 31 32 Binary Operations Binary Operations Definition Definition . A binary operation . A binary operation on a set S is on a set is commutative if and associative if only if a b b a ()()a b c a b c for all a,. b S for all a,,. b c S 33 34 Binary Operations Binary Operations Examples Examples . The binary operation defined by . The binary operation defined by a b a b is commutative and associative in . is commutative and associative in . The binary operation defined by a b ab is commutative and associative in 35 36 6 12/16/2018 Binary Operations Binary Operations . The binary operation defined by a b a b . The binary operation defined by is not commutative in . is not commutative in . The binary operation given by is not associative in 37 38 Binary Operations Group Theory . The binary operation defined by is not commutative in Bijective Maps . The binary operation given by is not associative in . For instance, (a b ) c (4 7) 2 5 but a( b c ) 4 (7 2) 1. 39 40 Bijective Maps Bijective Maps Definition Definition . A function f : X Y is . A function is called injective or one-to- called injective or one-to- one if one if f x1 f x 2 x 1 x 2. or x1 x 2 f x 1 f x 2 . 41 42 7 12/16/2018 Bijective Maps Bijective Maps Definition Definition . A function is . A function is called surjective or onto if called surjective or onto if for any yY , there exists for any , there exists xX with y f () x . with . i.e. if the image fx () is the whole set Y . 43 44 Bijective Maps Bijective Maps Definition Example . A bijective function or one- f: , f ( x ) 10x to-one correspondence is a function that is both injective and surjective. 45 46 Bijective Maps Bijective Maps Example Example f: X Y f( x ) f ( y ) 10xy 10 x y Therefore, f is one-to-one. Therefore, is one-to-one. If r , then log 10 r such that log10 r f(log10 r ) 10 r. 47 48 8 12/16/2018 Bijective Maps Bijective Maps Example Example Therefore, is one-to-one. Therefore, is one-to-one. If , then such that If , then such that . It implies that is onto. It implies that is onto. Hence is bijective. 49 50 Bijective Maps Bijective Maps Example Example f: , f ( m ) 3 m f: , f ( x ) 10x f( m ) f ( n ) 3 m 3 n m n Therefore, is one-to-one. 51 52 Bijective Maps Bijective Maps Example Example f:,() f x x2. f( x ) f ( y ) 10xy 10 x y Therefore, f is one-to-one. We assume that m is the pre-image of 4 , r log10 r log10 r then f ( m ) 3 m 4 m 4 / 3 . f(log10 r ) 10 r It implies that is not onto.