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Sets Part II: Operations

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The continuation of the first part.

The continuation of the first part.

Published in: Economy & Finance, Technology

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  • 1. SETS AND THE REAL NUMBER SYSTEM A Crash Course for Algebra Dummies
  • 2. Lesson Two: Set Operations How can something so simple be so f*cking complicated? This is like addition and subtraction, just different terms used.
  • 3. Recap of Defined Terms A SET is a well-defined collection of objects.  The members or objects are called  ELEMENTS. F = {cunt, vagina, pussy} A UNIVERSAL SET is the set of all elements  in a particular discussion. The CARDINALITY of a set states the number  of elements a set contains. Denoted by n(A). EQUAL SETS are sets with the exact same  elements.
  • 4. Set Unions Two sets can be quot;addedquot; together. The UNION  of two sets, denoted by ∪ , is the set of all elements either in the first set, OR the other. {1, 2} ∪ {red} = {1, 2, red} {x, y, z} ∪ {1, 2} = {x, y, z, 1, 2} The UNION of two sets is just a combination of  elements from both sets. Thus, A ∪ B is a combination of the elements from both sets A and B.
  • 5. Basic Properties of Unions A ∪ B = B ∪ A. Commutative Property of 1. Unions. A ∪ (B ∪ C) = (A ∪ B) ∪ C. Associative 2. Property of Unions. A ⊆ (A ∪ B). Obviously, since A ∪ B contains 3. A. A ∪ A = A. Uniting equal sets would result in 4. the same set. A ∪ ∅ = A. Identity Property of Unions. 5. A ⊆ B if and only if A ∪ B = B. In this case, A 6. = B, meaning A is an improper subset of B.
  • 6. Set Intersections The INTERSECTION of two sets, denoted by  ∩ , is the set of elements that are members of the first set AND the second set. {1, 2} ∩ {red, white} = ∅. {p, u, s, y} ∩ {f, u, c, k} = {u}. The INTERSECTION of two sets is just getting  the common elements from both sets. Thus, A ∩ B is a set of elements found in A and also found in B.
  • 7. Basic Properties of Intersections A ∩ B = B ∩ A. Commutative Property of 1. Intersections. A ∩ (B ∩ C) = (A ∩ B) ∩ C. Associative 2. Property of Intersections. A ∩ B ⊆ A. True, since A has a part that B has. 3. A ∩ A = A. Same elements from both sets. 4. A ∩ ∅ = ∅. Zero Property of Intersections. 5. A ⊆ B if and only if A ∩ B = A. In this case, A 6. = B, following from the 3rd property of intersections.
  • 8. Exercise 1: Unions & Intersections Given the following sets: F = {s, t, r, a, i, g, h} U = {g, a, y} C = {l, e, s, b, i, a, n} K = Ø or { } Find the following sets: 1. F ∪ U 2. C ∪ K 3. F ∪ U ∪ C 4. U ∩ K 5. F ∩ C 6. F ∩ U ∩ C 7. (F ∩ C) ∪ (U ∪ K) 8. (C ∪ U) ∩ (F ∩ K)
  • 9. Answers 1: Unions & Intersections F ∪ U = {s, t, r, a, i, g, h, y} 1. C ∪ K = {l, e, s, b, i, a, n} 2. F ∪ U ∪ C = {s, t, r, a, i, g, h, 3. F = {s, t, r, a, i, g, h} y, l, e, b, n} U = {g, a, y} U∩K=Ø 4. C = {l, e, s, b, i, a, n} F ∩ C = {s, a, i} 5. K = Ø or { } F ∩ U ∩ C = {a} 6. (F ∩ C) ∪ (U ∪ K) = {s, a, i, g, 7. y} (C ∪ U) ∩ (F ∩ K) = Ø 8.
  • 10. Set Differences Two sets can be “subtractedquot; too. The  DIFFERENCE of two sets, denoted by , is the set of all elements left when the other elements from the other set are removed. {1, 2} {red, white} = {1, 2}. {1, 2, green} {green} = {1, 2}. {1, 2, 3, 4} {1, 3} = {2, 4}.
  • 11. Set Differences In DIFFERENCES, the trick is to remove all the  common elements of the first set and the second set, then get the first set. A = {1, 2, 3, 4} and B = {1, 3, 5} AB {1, 2, 3, 4}{1, 3, 5} {2, 4}{5}  AB = {2, 4}
  • 12. Set Complements The COMPLEMENT of a set is the set of all  elements that are not included in the set, but are included in the universal set. It is denoted by an apostrophe (‘). Given that: U = {RE5, SH5, DMC4, GH:M}. If A = {RE5, SH5}, Then A’ = {DMC4, GH:M}.
  • 13. Basic Properties of Complements The trick in COMPLEMENTS is to get the  unmentioned elements in a set to form the new one. (A’)’ = A. Involution Property of Unions. 1. U’= Ø. All elements are included in the 2. universal set. Under the same premise, Ø’ = U. 3. A ∪ A’ = U. A’ is all members not part of A in 4. the universal set. Under the same premise, A ∩ A’ = Ø. 5. AA = Ø. Obvious, since these are equal sets. 6.
  • 14. Complements & Differences Given the following sets: U = {m, o, t, h, f, u, c, k, e, r} Q = {f, u, c, k} P = {r, o, t, c, h} E = {m, e, t, h} D = Ø or { } 5. (P ∪ Q)’ ∪ E 1. Q P Find the following 6. (PE) ∩ Q sets: 2. (EP)Q 3. (PQ) ∩ E 7. (P ∪ E ∪ Q)’ 8. (D ∩ U)’ ∪ Q 4. Q’P
  • 15. Complements and Differences Q P = {f, u, k} U = {m, o, t, h, f, u, c, k, e, r} 1. (EP)Q = {m, e} 2. P = {r, o, t, c, h} (PQ) ∩ E ={t, h} E = {m, e, t, h} Q = {f, u, c, k} 3. D = Ø or { } Q’P ={m, e, r} 4. (P ∪ Q)’ ∪ E = {m, e, t, h} or 5. E (PE) ∩ Q = {c} 6. (P ∪ E ∪ Q)’ = Ø 7. (D ∩ U)’ ∪ Q = 8. {m, o, t, h, f, u, c, k, e, r} or
  • 16. Cartesian Products {1, 2} × {red, white} = {(1, red), (1, white), (2, red), ( 2, white)}. A CARTESIAN PRODUCT is the set of all  ordered pairs from the elements of both sets. Denoted by ×. {1, 2, green} × {red, white, green} = {(1, red), (1, white), (1, green), (2, re d), (2, white), (2, green), (green, red
  • 17. Cartesian Products The trick behind CARTESIAN PRODUCTS is  to list down all possible pairs of elements such that the first element is from the first set and the second element is from the second set. A × ∅ = ∅. Zero Property of Products. 1. A × (B ∪ C) = (A × B) ∪ (A × C). Distribution. 2. (A ∪ B) × C = (A × C) ∪ (B × C). Distribution. 3. The CARDINALITY of Cartesian Products is  n(A) × n(B), wherein A and B are the given sets.
  • 18. Quiz Two: Set Operations For every slide you are given seven minutes to answer. Don’t cheat or I’ll kick your ass. Point system varies per question difficulty.
  • 19. True or False (One-Point Items) The cardinality of Cartesian Products is n(A) 1. × n(B), wherein A and B are the given sets. Uniting equal sets would result in a new set. 2. (A’)’ = A. 3. The intersection of two sets is a subset of 4. both sets. If Z = {negative numbers} then Z’ = 5. {nonnegative numbers}. AA = U. 6. The intersection of two sets always has a 7.
  • 20. Set-Building (One-Point Items) Given the following sets: U = {m, o, n, s, t, e, r, h, u} S = {h, u, n, t, s, m, e, n} R = {r, o, u, t, e} V = {r, e, m, o, t, e} H = Ø Find the following sets: (S ∩ V)(R ∩ S) 1. VR 5. (S ∪ V)R (H ∪ S’) ∩ R 2. 6. R ∩ V’ (V ∩ R ∩ S)’ 3. 7. U’ ∪ (SV) H’R 4. 8.
  • 21. Analysis (Two-Point Items) Write the power set of set P = {x, y}. After that, create a CARTESIAN PRODUCT between set P and its power set. Given that W is the set of days in a week, and M is the set of months in a year. Give the CARDINALITY of the Cartesian Products, and the Union of the Sets.