I am studying for a final by reproving theorems using logically equivalent forms. But I am having a little trouble negating. Consider the following,
Suppose that V is an n-dimensional vector space over a field F and that ${v_1, . . ., v_m}$ is a linearly independent set in V. Then $m <= n $ and there exists vectors $v_{m+1}, . . ., v_n$ such that ${v_1, . . ., v_m, v_{m+1}, . . . v_n}$ is a basis for V.
The theorem is stated using this logical flow: A_1 AND A_2 => B_1 AND B_2 where
A_1 = "Suppose that V is an n-dimensional vector space over a field F"
A_2 = "${v_1, . . ., v_m}$ is a linearly independent set in V"
B_1 = "$m<=n$"
B_2 = "there exists vectors $v_{m+1}, . . ., v_n$ such that ${v_1, ... ,v_m, v_{m+1}, ... v_n}$ is a basis for V"
We reformulate this as the following
A_1 AND A_2 => B_1 AND B_2 is the same as,
By currying, A_1 => (A_2 => B_1 AND B_2) this is the same as, taking the contrapositive, A_1 => (NOT B_1 AND NOT B_2 => NOT A_2)
Where:
A_1 = "Suppose that V is an n-dimensional vector space over a field F"
NOT A_2 = "${v_1, . . ., v_m}$ is a linearly dependent set in V"
NOT B_1 "$m > n$"
NOT B_2 "for all vectors $v_{m+1}, . . ., v_n$ such that ${v_1, . . ., v_m, v_{m+1}, . . . v_n}$ is a NOT a basis for V"
So we get:
Suppose that V is an n-dimensional vector space over a field F, Then $m > n$ OR for all vectors $v_{m+1}, . . ., v_n$ such that ${v_1, ..., v_m, v_m+1, . . . v_n}$ is a NOT a basis for V Implies ${v_1, . . ., v_m}$ is a linearly dependent set in V.
So for example let V be a 2-dimensional vector space over the field $R^2$,
Then 3 > 2 OR for all vectors $(x_4,y_4), (x_3,y_3), (x_2, y_2)$ such
that ${(x_1, y_2), (x_2, y_2) , (x_3,y_3), (x_4,y_4), (x_3,y_3), (x_2, y_2) }$ is a NOT a basis for V ( Is this the correct way to interpret $v_{m+1}, . . ., v_n$, since now $m > n$ ?? ) Implies ${(x_1, y_2), (x_2, y_2) , (x_3,y_3)}$ is a linearly dependent set in V.
So in this case: $3 > 2$ OR for all vectors $(x_4,y_4), (x_3,y_3), (x_2, y_2)$ such that
${(x_1, y_2), (x_2, y_2) , (x_3,y_3), (x_4,y_4), (x_3,y_3), (x_2,y_2) }$ IS A BASIS for V
Implies ${(x_1, y_2), (x_2, y_2) , (x_3,y_3)}$ is a linearly dependent set in V?
I am not sure what is the correct mathematical usage of the OR in this case.
Am I on the right track? Thank you for all your help.
