Determine whether each statement is true or false. If the statement is true, prove it. If the statement is false, provide a counter-example or other justification. The set of positive real mumbers is dense in R. For any real numbers a and b with a < b, the open interval (a, b) is dense in R.

List  the  members  of  the  set.  {x|x  is  an  odd  positive  integer  less  than  12}

Please make it clear and short

Without proving anything, determine if the following statements are true or false. For any false statements, give an counterexample. (a) A finite, nonempty set of real numbers always contains its supremum. (b) If a < L for every element a in the set A, then sup A < L. (c) If A and B are sets with the property that a < b for every a E A and b e B, then sup A < inf B. (d) If sup A = s and sup B = t, then sup(A+ B) = s+t. Here and elsewhere, the set A+B is defined as A+B = {a+b:a € A, and b e B}. (e) If sup A < sup B then there is an element of B that is an upper bound for A.

Suppose A is a set of real numbers such that 7EA and sup A >7. Must A contain a number grater than 7 ? Either prove that it must or give a caunter-example.