By Mark Dugopolski
Algebra for students, 4th variation, is designed to supply scholars with the algebra historical past wanted for additional college-level arithmetic classes. The unifying subject of this article is the advance of the abilities invaluable for fixing equations and inequalities, by means of the applying of these abilities to fixing utilized difficulties. this article comprises 2 chapters, Polynomial & Rational services, and Counting & chance, as well as these present in Dugopolski's Intermediate Algebra.
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Additional resources for Algebra for College Students, 5th Edition
So A ʝ B ϭ ͕2, 3͖. 3 E X A M P L E 4 A Venn diagram can be used to illustrate the result of Example 3(a). Since 2 and 3 belong to both A and B, they are placed in the overlapping region of Fig. 3. Since 0 is in A but not in B, it is placed inside the circle for A but outside B. Likewise 7 is placed inside B but outside A. Membership and equality Let A ϭ ͕1, 2, 3, 5͖ and B ϭ ͕2, 3, 7, 8͖. Place one of the symbols ϭ, blank to make each statement correct. a) 5 _____ A ʜ B b) 5 _____ A ʝ B c) A ʜ B _____ ͕l, 2, 3, 5, 7, 8͖ d) A ʝ B _____ ͕2͖ , ʦ, or in the Solution a) 5 ʦ A ʜ B because 5 is a member of A.
5 B A 1, 3 2 4 6, 8 5 Every possibility for membership in three sets is shown in the Venn diagram in Fig. 5. 6 shows the numbers from the three sets of Example 6 in the appropriate regions of this diagram. Since no number belongs to all three sets, there is no number in the center region of Fig. 6. Since 1 is in A, but is not in B or C it is placed inside circle A but outside circles B and C. Since 2 is in A and B, but is not in C, it is placed in the intersection of A and B, but outside C. Check that the remaining numbers are in the appropriate regions.
In Example 6, notice that different results are obtained from different placements of the parentheses. E X A M P L E 6 Operations with three sets Let A ϭ ͕1, 2, 3, 4͖, B ϭ ͕2, 5, 6, 8͖, and C ϭ ͕4, 5, 7͖. List the elements of each of these sets. qxd 10/31/07 6:37 PM Page 6 6 1-6 Chapter 1 The Real Numbers Solution a) The parentheses indicate that the union of A and B is to be found first and then the result, A ʜ B, is to be intersected with C. A ʜ B ϭ ͕1, 2, 3, 4, 5, 6, 8͖ Now examine A ʜ B and C to find the elements that belong to both sets: A ʜ B ϭ ͕1, 2, 3, 4, 5, 6, 8͖ C ϭ ͕4, 5, 7͖ The only numbers that are members of A ʜ B and C are 4 and 5.
Algebra for College Students, 5th Edition by Mark Dugopolski