MATH.20:4.4 - Propagate the comparison through the needed operations

For an order-preserving map F, L≤q≤U implies F(L)≤F(q)≤F(U). For an order-reversing map, the endpoints exchange roles. Establish the relevant behavior on the admitted domain.

For example, multiplication by a nonnegative scalar preserves numerical order, while multiplication by a negative one reverses it. Squaring is increasing on nonnegative inputs; an interval crossing zero needs its minimum and maximum square computed on that interval. From -2≤x≤1, the bound is 0≤x²≤4.

For sets, image and preimage under a fixed function preserve inclusion; complement reverses it in a fixed ambient set. Thus an enclosure can be carried into a subsequent transformation without requiring the original set to be enumerated.

Keep shared variables and constraints when combining bounds. If x∈[0,1], separately bounding x and -x gives -1≤x+(-x)≤1, but retaining the dependency gives equality to zero. The loose result is valid; its loss comes from forgetting that the terms use the same x.

A bound computed with approximate arithmetic must preserve the claimed direction after that computation. A computational continuation can use error bounds or enclosing arithmetic for this purpose; the mathematical argument states the comparison it must maintain.