12. Word the mathematical symbols using the introductory

words.

1) (VkK) (P(k) → Q(k)) for every k belonging to K capital P of k implies Q of k. 10. 2) kЄΦk k doesn't belong to the empty subset of K. 3) A’ A the set A prime is contained in the set A. 4) A A' the set A

contains the set A prime.5) A A' the union of A and A prime. 6) A∩A'

the intersection of A and A prime. 7) A ─ B the difference of A and B.

Text 2

1. Scan the text and give its main idea.

Statics. Elasticity

It may be assumed that a body consists of small particles or molecules, between which forces are acting. These molecular forces resist the change in the form of the body which external forces tend to produce. If such external forces are applied to the body, its particles are displaced and the mutual displacements continue until equilibrium is established between the external and internal forces. It is said in such a case that the body is in a state of deformation or strain. During

deformation the external forces acting upon the body do work, and this work is transformed completely or partially into the potential energy of strain. An example of such an accumulation of potential energy in a strained body is the case of a watch spring. If the forces which pro­duced the deformation of the body are now gradually diminished, the body returns wholly or partly to its original shape and during this reversed deformation the potential energy of strain, accumulated in the body, may be recovered in the form of external work.

The property of bodies of returning, after unloading, to their initial

form is called elasticity. It is said that the body is perfectly elastic if it

recovers its original shape completely after unloading; it is partially elastic if the deformation, produced by the external forces, does not disappear completely after unloading. In the case of a perfectly elastic body, the work done by the external forces during deformation will be completely transformed into the potential energy of strain. In the case of a partially elastic body, part of the work done by the external forces during deformation will be dissipated in the form of heat which will be developed in the body during non-elastic deformation. Experiments show that such structural materials as steel, wood, and stone may be considered as perfectly elastic within certain limits.

Assuming that the external forces acting upon the structure are known, it is a fundamental problem for the designer to establish such proportions of the members of the structure that it will improve the condition of a pertectly elastic body under all service conditions. Only under such conditions will there be continued reliable service from the structure and no permanent set in its members.

Vocabulary

аssume зробити припущення

external зовнішній

іnternal внутрішній

displasement зміщення

property властивість

recover відновити

2. Try to explain and prove why it is a problem for the designer to improve the condition of a pertectly elastic body under all service conditions.

3. Organize an international conference on the properties of

materials and problems of designers dealing with them. Use interpreters.

it may be assumed that, assuming that, assumed that, under the

assumption that, if, suppose if, if we consider, under con­sideration, in case, in such a case, then, provided that, it is said that, so, thus, though, therefore.

4. Introduce your questions about the mathematical notions below with:

can / could you / are you able to / may / might / allow me / must / we have to / we are to / we should / shall I / we ought to / I dare say / will you / would you

1) the given, 2) the conditions, 3) the ways to solve the problem given above / to find different values, 4) the results of the calculations.

5. Describe: a drawing, figure, scheme, project or design.

the drawing shows, side view, back view, front view, cross section, circle, straight line, the distance, as is shown at the figure, drop a perpendicular, pyramid base, slant height, base area, surface area, solve a

problem, obtain a result, given below, find a solution to / of the problem (calculate the volume, mathematical value), carry out calculations, rearrange the expression, sine, cosine, tangence, cotangens, triangle, pyramid, parallelogram, parallelepiped, rectangle, cone, cube, circle, oval, cylinder.