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

By Ilijas Farah

This ebook is meant for graduate scholars and study mathematicians drawn to set concept.

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Additional info for Analytic Quotients: Theory of Liftings for Quotients over Analytic Ideals on the Integers

Example text

11, the particle moves Y2 - Yi units in the y direction as it travels x2 - Xi units in the x direction. From ( 1 ) these quantities are related by xi) , (2) Y2 Yi = m ( X2 - - where m is the slope of the line. Equation (2) states that the movement in the y direction is proportional to the movement in the x direction and the slope m is the constant of proportionality. For this reason, m is said to measure the rate at which y changes with x along the line L. Example 9 In Example 7, we showed that the line determined by the points P1(4, 1) and P2(6, 9) has slope m = 4.

Let A, B, and C be the sets of points in Figure 1 . 17. Shade the following sets. (b) (A U B UC)' (a) A1 nB1 n c� (c) (A n B nO)' (d) [(A U B) n O]'. 10. Use a Venn diagram to verify that if A C B, then B' C A' . 11. Use a Venn diagram to verify the second DeMorgan law: (A n B) ' = A' U B'. 12. Use a Venn diagram to verify the following DeMorgan laws for three sets A, B, and C: ( A u B u C) ' = A' n B' n C' (A n B n C) ' = A ' u B' u C'. 13. (a) Conjecture DeMorgan laws for four sets A, B, C, and D.

B) A = { x I x is a real number satisfying x2 < O } and B = { x I x is a real number satisfying x2 = 4} . 3. If A and B are disjoint sets such that n(A U B) = n(A) , what can we say about B? 4. Verify the equation n(A U B) = n(A) + n(B) - n(A nB) for the following sets. (a) A = { a, c, e, g, i, kl, B = { e, f, g, h , i, i } (b) A = {x I is a positive integer < 10} B = Ix I x is an integer satisfying 1 � x � 7 } . (c) A = { Xerox, GE, ATT, IBM, Polaroid} B = { Sperry Rand, Kodak, Avco, Xerox, GEi.