Is the following statement true or false?  For each positive integer , either or  or .

What is ?

Is ?

Does 9 divide 29?

Suppose you know the following about a statement .

• , and are all true.
• is false.
• For all integers , if is true then is true.

What is the smallest integer for which you can be sure that is true for all integers ?

When attempting to prove by simple induction that for all integers , the base step would consist of proving which of the following inequalities?

Consider the sentences and . Are these sentences true or false under the interpretation where and range over the integers and is " is odd"?