Logic for analysis: proofs

Consider the statement

and the sequence defined by _=(-1) .

(a) Prove that the statement is true for =1 and for =-1.

(b) Prove that the statement is false for all other values of .

(c ) Do the same (i.e. prove that the statement is true for =1 and =-1 and false for all other values of ) for the sequence _{=1 +(-1)} .

In the integers, the next number after 3 is said to be 4 as 4 comes after 3 and there are no numbers between 3 and 4.

(a) Write down a formal statement that says that 4 is the next number after 3. You may use logical symbols, , and ,,, but you may not use anything else. In particular, you may not use addition or multiplication in your statement.

(b) Write down the statement that every number in has a next number in . Prove your statement. (As in Sheet 2, exercise 1 you should concentrate on the structure of your proof and you may write down as obvious any basic fact about that you need.)

(c) Write down the statement that no number in has a next number in , and prove this too.

(d) For the set = (0,1) , show that some elements of have a next number in and some elements of have no next number in .

(e) For each of your statements in (b) and (c), decide if it is true for the rational numbers, , instead of or , and justify your answer.