# Analytic Quotients: Theory of Liftings for Quotients over by Ilijas Farah By Ilijas Farah

This e-book is meant for graduate scholars and learn mathematicians attracted to set thought.

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Use a Venn diagram to verify the second DeMorgan law: (A n £ ) ' = A ' U ß ' . 12. Use a Venn diagram to verify the following DeMorgan laws for three sets A, By and C: (AU5uc)'= i ' n s ' n c ' ( i n ß n c ) ' = A'UB'UC". 13. (a) Conjecture DeMorgan laws for four sets A, B, C, and D. (b) Conjecture DeMorgan laws for n sets Ai, A2, . . , An. 14. Explain why the following are true. (a) (Af)'= A (b) AUA'=U (e) A fi A ' = 0 . 23. 23 *w 3 0 / 1 : SET THEORY 15. In each case determine which of the points x, y, z, w belong to the indicated set.

17. 18. (b) An#nc" (d) i ' n s ' n c . Follow the directions of Exercise 15 for the sets (a) A'VB (b) J5'UC" (c) A ' U t f ' U C " (d) A'UBUC'. In each part, find the values of x and y for which the given ordered pairs of integers are equal. (a) (*,7) = (3,7) (b) (2z,3) = (6,*/) (c) (4, y + 7) = (2s + 2, 14) (d) (x2, 9) = (16, 9 ) . Let A = {a, b, c, d) and B = {0, 1, 2}. (a) List the elements in A X B. (b) List the elements in B X A. (c) List the elements in A X A. (d) List the elements in B X B.

Rivers the following data were reported: 520 335 425 100 180 150 28 were polluted by sulfur compounds were polluted by phosphates were polluted by crude oil were polluted both by crude oil and sulfur compounds were polluted both by sulfur compounds and phosphates were polluted both by phosphates and crude oil were polluted by sulfur compounds, phosphates, and crude oil. Use this information to answer the following: (a) How many of the rivers were polluted by at least one of the three impurities?