Suppose that v = (v1, v2, ..., vn) and ū = (u₁, U2, ..., Un) are a pair of n-dimensional vectors. Assume that each component of the vector is a real number, so v and u are both members of the Set R¹ We will say that 7 and ū are "almost the same" when every component of is close to every component of ū. That is, v₁ is close to u₁, v2 is close to u2, etc (practically speaking, "close" means that their absolute difference is small). Assume that we are given the predefined predicate CloseTo(x, y) and the integer constant n. Use them to write a formal definition of the new predicate Almost The Same (7, ū) which asserts that ʼn dimensional vector is almost the same as ū.
Suppose that v = (v1, v2, ..., vn) and ū = (u₁, U2, ..., Un) are a pair of n-dimensional vectors. Assume that each component of the vector is a real number, so v and u are both members of the Set R¹ We will say that 7 and ū are "almost the same" when every component of is close to every component of ū. That is, v₁ is close to u₁, v2 is close to u2, etc (practically speaking, "close" means that their absolute difference is small). Assume that we are given the predefined predicate CloseTo(x, y) and the integer constant n. Use them to write a formal definition of the new predicate Almost The Same (7, ū) which asserts that ʼn dimensional vector is almost the same as ū.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Suppose that v = (v1, v2, ..., vn) and û = (u₁, U2, ..., Un) are a pair of n-dimensional vectors.
Assume that each component of the vector is a real number, so 7 and u are both members of the
set R¹.
We will say that and u are "almost the same" when every component of is close to every
component of ū. That is, v₁ is close to u₁, v2 is close to u2, etc (practically speaking, "close" means
that their absolute difference is small).
Assume that we are given the predefined predicate CloseTo(x, y) and the integer constant n.
Use them to write a formal definition of the new predicate Almost The Same (7, u) which
asserts that n dimensional vector is almost the same as ū.
Tip: It is not legal to say i v to refer to a component of v, because is not a set. Instead, use vi to
refer to the ith component of v. What set would i belong to in this case?
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