Sentence examples for mappings need not from inspiring English sources

Said mappings need not be one-to-one.

However, a weakly commuting mappings need not be commuting.

However, weakly uniformly contraction mappings need not be weakly commuting.

Note that, two weakly increasing mappings need not be nondecreasing.

Remark 3.2 Note that two weakly increasing mappings need not be nondecreasing.

It may be noted that semi-compatible mappings need not be compatible mappings.

It was then natural to ask if there exist weaker contractive conditions which do not imply the continuity of T. In 1968, this question was answered in confirmation by Kannan [20], who extended Theorem 1.1 to mappings that need not be continuous on X (but are continuous at their fixed point, see [21]).

Any pair of compatible as well as non-compatible self-mappings of a metric space (X, d) satisfies the property (E.A), but a pair of mappings satisfying the property (E.A) need not be non-compatible (see Example 1 of [27]).

Those mappings need to be supported by tools.

That is, intervention of a human expert is required, and the mappings need to be maintained.

None of two weakly increasing mappings need be non-decreasing.