We deal with a non–empty set of functions and a non– empty set of functions from a set A to a non–empty set B. In the case when B is a non–empty set, B is redefined. It yields a non–empty set of functions from A to B. An element of such a set is redefined as a function from A to B. Some theorems concerning these concepts are proved, as well as a number of schemes dealing with infinity and the A...

(2) Let N be a non empty set with non empty elements, and let S be a definite AMI over N , and let s be a state of S, and let i be an instructionlocation of S. Then s(i) is an instruction of S. (3) Let N be a non empty set with non empty elements, and let S be an AMI over N , and let s be a state of S. Then the instruction locations of S ⊆ dom s. (4) Let N be a non empty set with non empty elem...

One can prove the following proposition (1) For all topological spaces S, T holds Ω[: S, T :] = [:ΩS , ΩT :]. Let X be a set and let Y be an empty set. Note that [:X, Y :] is empty. Let X be an empty set and let Y be a set. Observe that [:X, Y :] is empty. We now state the proposition (2) Let X, Y be non empty topological spaces and x be a point of X. Then Y 7−→ x is a continuous map from Y int...

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In this paper x denotes a set and R denotes a non empty poset. Next we state two propositions: (1) For all order sorted sets X, Y of R holds X ∩Y is an order sorted set of R. (2) For all order sorted sets X, Y of R holds X ∪Y is an order sorted set of R. Let R be a non empty poset and let M be an order sorted set of R. A many sorted subset indexed by M is said to be an order sorted subset of M ...

For simplicity, we adopt the following rules: x, X are sets, i, n, m are natural numbers, r, s are real numbers, c, c1, c2, d are complex numbers, f , g are complex-valued functions, g1 is an n-element complex-valued finite sequence, f1 is an n-element real-valued finite sequence, T is a non empty topological space, and p is an element of En T. Let R be a binary relation and let X be an empty s...