MATH 32A: MIDTERM 1 REVIEW 1. Vectors 1. Let v = . a
... Solution: Three points determine the plane containing them, except in the rare case that they happen to be collinear. In this problem P is not on the line x = y = z, so we’re free to pick two points on the line. For simplicity, take Q = (0, 0, 0) and R = (1, 1, 1) (of course any other two points on ...
... Solution: Three points determine the plane containing them, except in the rare case that they happen to be collinear. In this problem P is not on the line x = y = z, so we’re free to pick two points on the line. For simplicity, take Q = (0, 0, 0) and R = (1, 1, 1) (of course any other two points on ...
§B. Appendix B. Topological vector spaces
... and we shall also use the notation B(x, r) = {y ∈ X | d(x, y) ≤ r} for the closed ball. Such a metric need not be translation invariant, but it will usually be so in the cases we consider; translation invariance (also just called invariance) here means that d(x + a, y + a) = d(x, y) for x, y, a ∈ X ...
... and we shall also use the notation B(x, r) = {y ∈ X | d(x, y) ≤ r} for the closed ball. Such a metric need not be translation invariant, but it will usually be so in the cases we consider; translation invariance (also just called invariance) here means that d(x + a, y + a) = d(x, y) for x, y, a ∈ X ...
Interval-valued Fuzzy Vector Space
... Definition 5. Let F be an IVFS and x, y ∈ F where x = [a1 , b1 ] and y = [a2 , b2 ] , then x = y if and only if a1 = a2 and b1 = b2 . The logical operators ≤ and < are given in the following definitions. Definition 6. Let F be an IVFS and x, y ∈ F where x = [a1 , b1 ] and y = [a2 , b2 ] , then x ≤ y ...
... Definition 5. Let F be an IVFS and x, y ∈ F where x = [a1 , b1 ] and y = [a2 , b2 ] , then x = y if and only if a1 = a2 and b1 = b2 . The logical operators ≤ and < are given in the following definitions. Definition 6. Let F be an IVFS and x, y ∈ F where x = [a1 , b1 ] and y = [a2 , b2 ] , then x ≤ y ...
