...9 Let A be a subset of {1, 2, 11} such that |A| = 7. Prove that there exist ï, y ¤ A such that ï ‡ y and x + y = 12.

(b) Suppose that X,Y and Z are subsets of {1, 2, 3, . . . , 10} and |X| = |Y| = |Z| = 7.Deduce that X ∩ Y ∩ Z is non-empty. [Hint: Consider (X ∩ Y) ∪ Z.]

2.28 let xa be a translation of R Then (1) La is a continuous bijection from onto R (ii) The image of an open set under Da is an oper set. of (iii) let & be an open set. The component interval +a are exactly The images of the component intervals of the set & under translation a the goal of this section is to establish the following result.

Let S = {x | 6x – 1 E Q} and T be the - set of all odd integers. Prove that S and T have the same cardinality (no need to prove bijectivity of functions). Do not use the Schroder-Bernstein Theorem.