3. recall that p is a condensation point of s if each mrp contains uncountably many points of s. the set of condensation points of s is denoted as s∗. prove: (a) if s ⊂ t then s∗ ⊂ t∗. (b) (s ∪ t)∗ = s∗ ∪ t∗. (c) s∗ ⊂ s∗ where s∗ = (s)∗ (d) s∗ is closed in m; that is, s∗ ⊂ s∗ where s∗ = (s∗) . (e) s∗∗ ⊂ s∗ where s∗∗ = (s∗)∗. (f) calculate n∗, q∗, r∗, and (r q)∗. (g) give an example showing that s∗ can be a proper subset of (s)∗. thus, (c) is not in general an equality

The area of a rectangle is 54x^9y^8 square yards.if the length of the rectangle is 6x^3y^4 yards,which expression represents the width of the rectangle in yards

When the ball has traveled a distance of

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?