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The examples "is greater than", "is at least as great as", and "is equal to" are transitive relations on various sets. As are the set of real numbers or the set of natural numbers: whenever x > y and y > z, then also x > z whenever x ≥ y and y ≥ z, then also x ≥ z whenever x = y and y = z, then also x = z. More examples of transitive ...
is invertible, since the derivative f′(x) = 3x 2 + 1 is always positive. If the function f is differentiable on an interval I and f′(x) ≠ 0 for each x ∈ I, then the inverse f −1 is differentiable on f(I). [17] If y = f(x), the derivative of the inverse is given by the inverse function theorem,
In the monoid of binary endorelations on a set (with the binary operation on relations being the composition of relations), the converse relation does not satisfy the definition of an inverse from group theory, that is, if is an arbitrary relation on , then does not equal the identity relation on in general.
Throughout this article, capital letters (such as ,,,,, and ) will denote sets.On the left hand side of an identity, typically, will be the leftmost set, will be the middle set, and
A negative correlation between variables is also called inverse correlation. Negative correlation can be seen geometrically when two normalized random vectors are viewed as points on a sphere, and the correlation between them is the cosine of the circular arc of separation of the points on a great circle of the sphere. [ 1 ]
The right side is equal to 1 and the chain rule can be applied to the left side: ... (for positive x) has inverse = . = ; = ... The inverse of the derivative of f(x) ...
This leads to the equivalent characterization: a matrix Q is orthogonal if its transpose is equal to its inverse: =, where Q −1 is the inverse of Q. An orthogonal matrix Q is necessarily invertible (with inverse Q −1 = Q T ), unitary ( Q −1 = Q ∗ ), where Q ∗ is the Hermitian adjoint ( conjugate transpose ) of Q , and therefore normal ...
In a vector space, the additive inverse −v (often called the opposite vector of v) has the same magnitude as v and but the opposite direction. [11] In modular arithmetic, the modular additive inverse of x is the number a such that a + x ≡ 0 (mod n) and always exists. For example, the inverse of 3 modulo 11 is 8, as 3 + 8 ≡ 0 (mod 11). [12]