Logic for analysis: statements

Exercise.

(a) Write down the negation of the following statements in such a way as there are no not signs in your answer. Logical symbols, , , , and are OK, but you must not use any other symbols such as +, etc.

A: ( )

B: ( ( ) )

(b) Now interpret the domain above in turn as (i) = , (ii) =0,1 , (iii) = . Say for each of A,B whether you think the original statement is true or false, and write a brief sentence in English indicating in an informal way why you think so.

(c) Attempt to write down proofs of your assertions in part (b). Note: your proof will ultimately rely on obvious properties of the order relation and other properties of the integers or the reals. You may take these as given. (Write obvious for the reason.) Your work will be marked on whether you have the correct structure for a proof. Use the following model answers for case (iii) as a model.

Answer for case (iii) = : Both statements are true, and proved as follows.

A: ( )

Proof.

Subproof.

Let be arbitrary

Subproof.

Let be =+31 .

Then +31= so . (Obvious.)

So .

B: ( ( ) )

Proof.

Subproof.

Let be arbitrary

Subproof.

Let be arbitrary.

Subproof.

Suppose .

Subproof.

Let be (+)/2 .

Then and as (obvious).

So ( ) .

Then ( ) .

So ( ) .

Exercise.

Consider the following statement about a sequence (₎ and the nearness of its values to the number two.

C: 0 _-2

(a) Write down the negation of this statement.

(b) Decide whether the statement C is true or false, and prove your assertion for ₌₁.