V = R 2 I . ∀ a ⃗ = ( x 1 , y 1 ) ∈ V , b ⃗ = ( x 2 , y 2 ) ∈ V a ⃗ + b ⃗ = ( x 1 , y 1 ) + ( x 2 , y 2 ) = ( 0 , y 1 + y 2 ) ∈ V I I . ∀ c ∈ R , ∀ a ⃗ = ( x 1 , y 1 ) ∈ V c ⋅ a ⃗ = c ⋅ ( x 1 , y 1 ) = ( 0 , c y 1 ) ∈ V V=R^2\\
I. \forall \vec{a}=(x_1,y_1)\in V, \vec{b}=(x_2,y_2)\in V\\
\vec{a}+\vec{b}=(x_1,y_1)+(x_2,y_2)=(0,y_1+y_2)\in V\\
II. \forall c\in R, \forall \vec{a}=(x_1,y_1)\in V\\
c\cdot\vec{a}=c\cdot(x_1,y_1)=(0,cy_1)\in V V = R 2 I .∀ a = ( x 1 , y 1 ) ∈ V , b = ( x 2 , y 2 ) ∈ V a + b = ( x 1 , y 1 ) + ( x 2 , y 2 ) = ( 0 , y 1 + y 2 ) ∈ V II .∀ c ∈ R , ∀ a = ( x 1 , y 1 ) ∈ V c ⋅ a = c ⋅ ( x 1 , y 1 ) = ( 0 , c y 1 ) ∈ V
Axioms
1 ) a ⃗ + b ⃗ = b ⃗ + a ⃗ a ⃗ + b ⃗ = ( x 1 , y 1 ) + ( x 2 , y 2 ) = ( 0 , y 1 + y 2 ) b ⃗ + a ⃗ = ( x 2 , y 2 ) + ( x 1 , y 1 ) = ( 0 , y 2 + y 1 ) = = ( 0 , y 1 + y 2 ) = a ⃗ + b ⃗ t r u e 1) \vec{a}+\vec{b}= \vec{b}+\vec{a}\\
\vec{a}+\vec{b}=(x_1,y_1)+(x_2,y_2)=(0,y_1+y_2)\\
\vec{b}+\vec{a}=(x_2,y_2)+(x_1,y_1)=(0, y_2+y_1)=\\
=(0,y_1+y_2)= \vec{a}+\vec{b}\\
true 1 ) a + b = b + a a + b = ( x 1 , y 1 ) + ( x 2 , y 2 ) = ( 0 , y 1 + y 2 ) b + a = ( x 2 , y 2 ) + ( x 1 , y 1 ) = ( 0 , y 2 + y 1 ) = = ( 0 , y 1 + y 2 ) = a + b t r u e
2. ∀ d ⃗ = ( x 3 , y 3 ) ∈ V ( a ⃗ + b ⃗ ) + d ⃗ = a ⃗ + ( b ⃗ + d ⃗ ) a ⃗ + ( b ⃗ + d ⃗ ) = ( x 1 , y 1 ) + ( 0 , y 2 + y 3 ) = = ( 0 , y 1 + y 2 + y 3 ) ( a ⃗ + b ⃗ ) + d ⃗ = ( 0 , y 1 + y 2 ) + ( x 3 , y 3 ) = = ( 0 , y 1 + y 2 + y 3 ) = a ⃗ + ( b ⃗ + c ⃗ ) t r u e 2. \forall \vec{d}=(x_3,y_3)\in V\\
(\vec{a}+\vec{b})+\vec{d}=\vec{a}+(\vec{b}+\vec{d})\\
\vec{a}+(\vec{b}+\vec{d})=(x_1,y_1)+(0, y_2+y_3)=\\
=(0, y_1+y_2+y_3)\\
(\vec{a}+\vec{b})+\vec{d}=(0, y_1+y_2)+(x_3,y_3)=\\
=(0, y_1+y_2+y_3)=\vec{a}+(\vec{b}+\vec{c})\\
true 2.∀ d = ( x 3 , y 3 ) ∈ V ( a + b ) + d = a + ( b + d ) a + ( b + d ) = ( x 1 , y 1 ) + ( 0 , y 2 + y 3 ) = = ( 0 , y 1 + y 2 + y 3 ) ( a + b ) + d = ( 0 , y 1 + y 2 ) + ( x 3 , y 3 ) = = ( 0 , y 1 + y 2 + y 3 ) = a + ( b + c ) t r u e
3. ∃ 0 ⃗ = ( x 2 , y 2 ) ∈ V , ∀ a ⃗ = ( x 1 , y 1 ) ∈ V a ⃗ + 0 ⃗ = a ⃗ a ⃗ + 0 ⃗ = ( x 1 , y 1 ) + ( x 2 , y 2 ) = ( 0 , y 1 + y 2 ) ≠ a ⃗ ∄ 0 ⃗ f a l s e 3. \exist\vec{0}=(x_2,y_2)\in V, \forall\vec{a}=(x_1,y_1)\in V\\
\vec{a}+\vec{0}=\vec{a}\\
\vec{a}+\vec{0}=(x_1,y_1)+(x_2,y_2)=(0, y_1+y_2)\neq\vec{a}\\
\not\exist\vec{0}\\
false 3.∃ 0 = ( x 2 , y 2 ) ∈ V , ∀ a = ( x 1 , y 1 ) ∈ V a + 0 = a a + 0 = ( x 1 , y 1 ) + ( x 2 , y 2 ) = ( 0 , y 1 + y 2 ) = a ∃ 0 f a l se
4. ∀ a ⃗ = ( x 1 , y 1 ) ∈ V ∃ b ⃗ = ( x 2 , y 2 ) ∈ V a ⃗ + b ⃗ = 0 ⃗ ∄ 0 ⃗ f a l s e 4. \forall\vec{a}=(x_1,y_1)\in V\exist\vec{b}=(x_2,y_2)\in V\\
\vec{a}+\vec{b}=\vec{0}\\
\not\exist\vec{0}\\
false 4.∀ a = ( x 1 , y 1 ) ∈ V ∃ b = ( x 2 , y 2 ) ∈ V a + b = 0 ∃ 0 f a l se
5. ∀ c , k ∈ R , ∀ a ⃗ = ( x 1 , y 1 ) ∈ V ( c + d ) ⋅ a ⃗ = c ⋅ a ⃗ + d ⋅ a ⃗ ( c + d ) ⋅ a ⃗ = ( c + d ) ⋅ ( x 1 , y 1 ) = ( 0 , ( c + d ) y 1 ) c ⋅ a ⃗ = c ⋅ ( x 1 , y 1 ) = ( 0 , c y 1 ) d ⋅ a ⃗ = d ⋅ ( x 1 , y 1 ) = ( 0 , d y 1 ) c ⋅ a ⃗ + d ⋅ a ⃗ = ( 0 , c y 1 ) + ( 0 , d y 1 ) = ( 0 , c y 1 + d y 1 ) = = ( c + d ) ⋅ a ⃗ t r u e 5. \forall c, k\in R, \forall \vec{a}=(x_1,y_1)\in V\\
(c+d)\cdot\vec{a}=c\cdot\vec{a}+d\cdot\vec{a}\\
(c+d)\cdot\vec{a}=(c+d)\cdot(x_1,y_1)=(0,(c+d)y_1)\\
c\cdot\vec{a}=c\cdot(x_1,y_1)=(0,cy_1)\\
d\cdot\vec{a}=d\cdot(x_1,y_1)=(0,dy_1)\\
c\cdot\vec{a}+d\cdot\vec{a}=(0,cy_1)+(0,dy_1)=(0,cy_1+dy_1)=\\
=(c+d)\cdot\vec{a}\\true 5.∀ c , k ∈ R , ∀ a = ( x 1 , y 1 ) ∈ V ( c + d ) ⋅ a = c ⋅ a + d ⋅ a ( c + d ) ⋅ a = ( c + d ) ⋅ ( x 1 , y 1 ) = ( 0 , ( c + d ) y 1 ) c ⋅ a = c ⋅ ( x 1 , y 1 ) = ( 0 , c y 1 ) d ⋅ a = d ⋅ ( x 1 , y 1 ) = ( 0 , d y 1 ) c ⋅ a + d ⋅ a = ( 0 , c y 1 ) + ( 0 , d y 1 ) = ( 0 , c y 1 + d y 1 ) = = ( c + d ) ⋅ a t r u e
6. ∀ c ∈ R , ∀ a ⃗ = ( x 1 , y 1 ) ∈ V , b ⃗ = ( x 2 , y 2 ) ∈ V c ⋅ ( a ⃗ + b ⃗ ) = c ⋅ a ⃗ + c ⋅ b ⃗ c ⋅ ( a ⃗ + b ⃗ ) = c ⋅ ( 0 , y 1 + y 2 ) = ( 0 , c ( y 1 + y 2 ) ) c ⋅ a ⃗ = ( 0 , c y 1 ) c ⋅ b ⃗ = ( 0 , c y 2 ) c ⋅ a ⃗ + c ⋅ b ⃗ = ( 0 , c y 1 ) + ( 0 , c y 2 ) = ( 0 , c y 1 + c y 2 ) = = c ⋅ ( a ⃗ + b ⃗ ) t r u e 6. \forall c\in R, \forall \vec{a}=(x_1,y_1)\in V,\\
\vec{b}=(x_2,y_2)\in V\\
c\cdot(\vec{a}+\vec{b})=c\cdot\vec{a}+c\cdot\vec{b}\\
c\cdot(\vec{a}+\vec{b})=c\cdot(0,y_1+y_2)=(0,c(y_1+y_2))\\
c\cdot\vec{a}=(0,cy_1)\\
c\cdot\vec{b}=(0,cy_2)\\
c\cdot\vec{a}+c\cdot\vec{b}=(0,cy_1)+(0,cy_2)=(0,cy_1+cy_2)=\\
=c\cdot(\vec{a}+\vec{b})\\true 6.∀ c ∈ R , ∀ a = ( x 1 , y 1 ) ∈ V , b = ( x 2 , y 2 ) ∈ V c ⋅ ( a + b ) = c ⋅ a + c ⋅ b c ⋅ ( a + b ) = c ⋅ ( 0 , y 1 + y 2 ) = ( 0 , c ( y 1 + y 2 )) c ⋅ a = ( 0 , c y 1 ) c ⋅ b = ( 0 , c y 2 ) c ⋅ a + c ⋅ b = ( 0 , c y 1 ) + ( 0 , c y 2 ) = ( 0 , c y 1 + c y 2 ) = = c ⋅ ( a + b ) t r u e
7. ∀ c , k ∈ R , ∀ a ⃗ = ( x 1 , y 1 ) ∈ V ( c ⋅ d ) ⋅ a ⃗ = c ⋅ ( d ⋅ a ⃗ ) ( c ⋅ d ) ⋅ a ⃗ = ( c ⋅ d ) ⋅ ( x 1 , y 1 ) = ( 0 , c ⋅ d y 1 ) d ⋅ a ⃗ = d ⋅ ( x 1 , y 1 ) = ( 0 , d y 1 ) c ⋅ ( d ⋅ a ⃗ ) = c ⋅ ( 0 , d y 1 ) = ( 0 , c d y 1 ) = ( c ⋅ d ) ⋅ a ⃗ t r u e 7. \forall c, k\in R, \forall \vec{a}=(x_1,y_1)\in V\\
(c\cdot d)\cdot\vec{a}=c\cdot(d\cdot\vec{a})\\
(c\cdot d)\cdot\vec{a}=(c\cdot d)\cdot(x_1,y_1)=(0,c\cdot dy_1)\\
d\cdot\vec{a}=d\cdot(x_1,y_1)=(0,dy_1)\\
c\cdot(d\cdot\vec{a})=c\cdot(0,dy_1)=(0,cdy_1)=
(c\cdot d)\cdot\vec{a}\\true 7.∀ c , k ∈ R , ∀ a = ( x 1 , y 1 ) ∈ V ( c ⋅ d ) ⋅ a = c ⋅ ( d ⋅ a ) ( c ⋅ d ) ⋅ a = ( c ⋅ d ) ⋅ ( x 1 , y 1 ) = ( 0 , c ⋅ d y 1 ) d ⋅ a = d ⋅ ( x 1 , y 1 ) = ( 0 , d y 1 ) c ⋅ ( d ⋅ a ) = c ⋅ ( 0 , d y 1 ) = ( 0 , c d y 1 ) = ( c ⋅ d ) ⋅ a t r u e
8. ∃ 1 ∈ R , ∀ a ⃗ = ( x 1 , y 1 ) ∈ V 1 ⋅ a ⃗ = a ⃗ 1 ⋅ a ⃗ = 1 ⋅ ( x 1 , y 1 ) = ( 0 , 1 ⋅ y 1 ) = ( 0 , y 1 ) ≠ a ⃗ f a l s e 8. \exist 1\in R, \forall\vec{a}=(x_1,y_1)\in V\\
1\cdot\vec{a}=\vec{a}\\
1\cdot\vec{a}=1\cdot(x_1,y_1)=(0,1\cdot y_1)=(0,y_1)\neq\vec{a}\\
false 8.∃1 ∈ R , ∀ a = ( x 1 , y 1 ) ∈ V 1 ⋅ a = a 1 ⋅ a = 1 ⋅ ( x 1 , y 1 ) = ( 0 , 1 ⋅ y 1 ) = ( 0 , y 1 ) = a f a l se
V V V 2
#340153 on Dec 2023
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