We say that a set of integers is sparse if any two elements in the set differ by at least 3. Find the number of sparse subsets of $\{1, 2, 3, \dots, 12\}.$ (Note that the empty set is also sparse.)

1- is the product of two rational numbers irrational or rational? first, make a hypothesis by multiplying two rational numbers. then, use variables such as x=a/b and y=c/d and the closure property of integers to prove your hypothesis. 2- what do you think the product of a nonzero rational number and an irrational number is? is it rational or irrational? make use of variables, the closure property of integers, and possibly a proof by contradiction to prove your hypothesis. 3- why do we have to specify that the rational number must be nonzero when we determine what the product of a nonzero rational number and an irrational number is? if the rational number were 0, would it give us the same result we found in part b?

Cone a has a surface area of 384 ft2 and cone b has a surface area of 96 ft? . what is the ratio of their volume?

Is the underlined participial phrase correctly placed or misplaced? i smelled each and every flower strolling through the garden. a. correctly placed participial phrase b. misplaced participial phrase