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In size theory, the natural pseudo-distance between two size pairs $$(M,\varphi:M\to \mathbb{R})$$, $$(N,\psi:N\to \mathbb{R})$$ is the value $$\inf_h |\varphi-\psi\circ h|_\infty$$, where $$h$$ varies in the set of all diffeomorphisms from the manifold $$M$$ to the manifold $$N$$ and $$\|\cdot\|_\infty$$ is the supremum norm. $$M$$, $$N$$ are assumed to be $$C^1$$ closed manifolds and the measuring functions $$\varphi,\psi$$ are assumed to be $$C^1$$. Put another way, the natural pseudo-distance measures the infimum of the change of the measuring function induced by a diffeomorphism $$h:M\to N$$.

Main properties
It can be proved For a fixed length n, the Hamming distance is a metric on the vector space of the words of that length, as it obviously fulfills the conditions of non-negativity, identity of indiscernibles and symmetry, and it can be shown easily by complete induction that it satisfies the triangle inequality as well. The Hamming distance between two words a and b can also be seen as the Hamming weight of a&minus;b for an appropriate choice of the &minus; operator.

For binary strings a and b the Hamming distance is equal to the number of ones in a xor b. The metric space of length-n binary strings, with the Hamming distance, is known as the Hamming cube; it is equivalent as a metric space to the set of distances between vertices in a hypercube graph. One can also view a binary string of length n as a vector in $$\mathbb{R}^n$$ by treating each symbol in the string as a real coordinate; with this embedding, the strings form the vertices of an n-dimensional hypercube, and the Hamming distance of the strings is equivalent to the Manhattan distance between the vertices.