Question: #4 Question 7. (Total 8pts) What the duplicate Q + Q: = {r, + 72 |r1,12 E Q} of the set of all rational numbers

Question 7. (Total 8pts) What the duplicate Q + Q: = {r, + 72 |r1,12 E Q} of the set of all rational numbers Q is compared to R? That is, find out what Q + Q is, and check whether Q + Q is a proper subset of R. What the duplicate I + I: = {s + t | s, te I} of the set of all irrational numbers I is compared to R? That is, find out what I + II is, and check whether I + I is a proper subset of R. Question 8. (Total 38pts) Recall the definition of the classical (triplicate) Cantor set C and show that: (1) The Cantor set C has a (linear) measure zero by computation; that is, C is of zero width. (2) The Cantor set C has as many elements as the entire real line R; that is, C is of continuum cardinality. (3) The Cantor set C has a fractional dimension log3 2; that is, C exists in a "fractional" world. (4) The Cantor set C is compact (use my definition of compact set given in class to simply the proof). (5) The Cantor set C is perfect; that is, although it is often called a "dust" set, C has no isolated point. (6) The Cantor set C is totally disconnected; that is, every pair of points of C are disconnected. (7) The Cantor set C is nowhere-dense; that is, C is composed entirely of discrete points. (8) The duplicate C + C of the Cantor set C covers the entire closed interval [0, 2]

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