Grammar revision.

Summarize all the grammar rules and the verb to do functions so far studied.

1. Imperative Sentences.

E.g. Suppose (Let us suppose) we have a theorem. Prove it deductively. Let her (him, them) do it.

2. Indefinite Tense-Aspect Forms.

Present	E. g. Mathematicians prove (must prove) theorems deductively and rigorously. Theorems are proved (must be proved) deductively and rigorously.
Past	E. g. Mathematicians proved (were to prove, had to prove) theorems deductively. Theorems were proved (were to be proved, had to be proved) deductively. A deductive proof is (was) much spoken and written about. A rigorous and elegant deductive proof is (was) looked at with admiration.

3. Questions.

E.g. Who proves (must prove) theorems? Who proved (was to prove, had to prove) theorems? What do (did) mathematicians do? How do (did) mathematicians prove theorems? Mathematicians prove(d) theorems, don't (didn't) they? Do (Did) mathematicians prove theorems deductively or inductively? What is a deductive proof? Is it difficult to prove theorems deductively?

4. Negations.

E.g. Don't prove theorems that way. Don't let him (her, them) prove theorems that way. Let us not prove theorems that way. Mathematicians do not (did not) prove theorems that way. No mathematician proves theorems that way. Don't (Didn't) mathematicians prove theorems deductively. Mathematicians prove(d) no theorem that way. Mathematicians prove(d) nothing that way. Mathematicians prove(d) theorems that way nowhere.

There is much thinking, reasoning, proving and justifying in maths. Is there? There is no arguing (not any argument) in this theory. There exists (emerged) a new proof of this theorem. Does there exist...? There does not exist... Did there emerge...? There did not emerge a new proof of this theorem.

5. Impersonal Sentences.

E.g. It is (pre)supposed that mathematicians prove(d) theorems that way. One (does not) suppose(s) (can hardly suppose)... We (you, they) must (not) suppose... People should (not) suppose...

6. Emphatic Sentences.

E.g. It is (was) mathematicians who prove(d) theorems. It is (was) deductively that mathematicians prove(d) theorems. Do prove theorems deductively! Mathematicians do (did) prove theorems deductively. Whatever (Whichever) Euclid's proof you take, it is deductive. The earlier you master the procedure of a deductive proof, the sooner you appreciate math rigour.

7. Noun Substitution.

E.g. The proof(s) by deduction is (are) much more rigorous than that of (those of) by induction. Deduction and rigour are essentials of a math proof. The former and the latter are essentials of a math proof. These proofs are valid but try to establish more rigorous ones.

8. Verb Substitution.

E.g. Mathematicians prove theorems inductively rather rarely but physicists do it regularly. Mathematicians prove what they do (= prove) deductively and rigorously.

9. The Verb To Do Functions.

E.g. 1. These students do maths. 2. What do these students do? 3. They do prove theorems. Do prove this theorem deductively! 4. They do not prove theorems but we do. They prove what they do deductively.

2. Study the text, find the verb to do and state its functions.

3. Underline the affixes, state what part of speech they indicate and translate them into Ukrainian:

incident – incidental; rational – rationale – rationalism – rationalist – rationalistic – rationality – rationalize – rationally; compartment – compartmentalization; add – addition – additive; concept – conception – conceptual.

4. Look through the text and give Ukrainian equivalents for the following words and word-combinations:

regular members; incidental members; mental discipline; science learning; associative, commutative and distributive properties; abstractions from experience; deductive reasoning; implication; connotation; compartmentalization; province.

5. Look through the text and find English equivalents for the following words and word-combinations:

детально; мати відношення до чогось; відкриття; набуті знання; значення; виводити (формулу); замінювати; точка зору; походження; галузь.

6. Write an appropriate word or word-combination in the following spaces:

1. The incidental members were called … .

2. The regular members were named … .

3. … , viewed as a whole, is a collection of branches.

4. The largest branch is that which builds on the ordinary whole numbers, fractions and irrational numbers, or what collectively is called … .

5. From the concepts and axioms … are … .

6. The basic … of the main branches of maths are … .

7. The concepts of a … relationship between variables, is almost totally a … .

concept; mental creation; function; the real number system; theorems; maths as a science; auditors; deduced; abstractions from experience; mathematicians.

7. Combine the words from the left- and right-hand columns to make word-combinations. Translate them into Ukrainian:

general	discipline
science	properties
mental	values
differential	terms
associative	creation
physical	maths
quantative	whole
geometrical	knowledge
human	world
objective	knowledge
pure	learning
scientific	equations
organic	counterparts

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