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

By Ilijas Farah

This booklet is meant for graduate scholars and learn mathematicians attracted to set thought.

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3 + 8 1 . 1 + 8 2 . 0 + 8 9 . 0 and ( 0 + 1 + 2 + 3 + 4 + 5 ) b + ( 0 + 1 + 4 + 9 + 1 6 + 25)m = 6 8 . 7 + 1 4 2 . 6 + 2 4 3 . 3 + 3 2 8 . 0 + 4 4 5 . 6. 18. Exercise 13 Let x = the number of meals prepared and sold in a week. 50x. To find the break-even point, we set R = C and solve for x. 25x = $1375, and x = $1100. 5 Results to remember: 1. When determining the region defined by an inequality, first draw the corresponding line, then choose a test point. a) Suppose the test point is above the line: (1) If the coordinates of the point satisfy the inequality, then the solution set is the half plane above the line.

18. Exercise 13 Let x = the number of meals prepared and sold in a week. 50x. To find the break-even point, we set R = C and solve for x. 25x = $1375, and x = $1100. 5 Results to remember: 1. When determining the region defined by an inequality, first draw the corresponding line, then choose a test point. a) Suppose the test point is above the line: (1) If the coordinates of the point satisfy the inequality, then the solution set is the half plane above the line. (2) If the coordinates of the point do not satisfy the inequality, then the solution set is the half plane below the line.

5 Exercise 9 Lines Inequalities 1. 4x+ 2y < 5. 4x + 2y = 5. 2. -x+ y < 0. -x + y = 0. y > 0. y = 0. 3. , £«, and £,. The student can easily plot these lines. ) We next determine the regions defined by each of the inequalities. will use the point (2,1) as a test point. 1. (2,1) lies above &-. We Inserting these values into inequality 1 yields 10 < 5, which is false. Therefore 2. inequality 1 defines the region below and on_ I*. (2,1) lies below %~. Inserting these values into inequality 2 yields -1 < 0, which is true.