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Understanding Set Operations: Union, Intersection, and More in Discrete Mathematics

This overview covers fundamental operations on sets, including union, intersection, complement, and symmetric difference in discrete mathematics. With practical examples and Venn diagrams, it illustrates how to effectively combine and manipulate sets to derive new sets. The principles of commutative, associative, and distributive properties are discussed, along with de Morgan's laws and the addition principle for determining the cardinality of unions. This resource is essential for students of discrete structures seeking to grasp the foundational concepts of set theory.

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Understanding Set Operations: Union, Intersection, and More in Discrete Mathematics

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  1. CSCI 1900Discrete Structures Operations on SetsReading: Kolman, Section 1.2

  2. Operation on Sets • An operation on a set is where two sets are combined to produce a third

  3. Union • AB = {x | x A or x B} • Example:Let A = {a, b, c, e, f} and B = {b, d, r, s}AB = {a, b, c, d, e, f, r, s} • Venn diagram

  4. Intersection • AB = {x | x A and x B} • Example: Let A = {a, b, c, e, f},B = {b, e, f, r, s}, and C = {a, t, u, v}.AB = {b, e, f}AC = {a}BC = { } • Venn diagram

  5. Disjoint Sets Disjoint sets are sets where the intersection results in the empty set Not disjoint Disjoint

  6. Unions and Intersections Across Multiple Sets Both intersection and union can be performed on multiple sets • ABC = {x | x A or x B or x C} • ABC = {x | x A and x B and x C} • Example:A = {1, 2, 3, 4, 5, 7}, B = {1, 3, 8, 9}, and C = {1, 3, 6, 8}.ABC = {1, 2, 3, 4, 5, 6, 7, 8, 9}ABC = {1, 3}

  7. Complement • The complement of A (with respect to the universal set U) – all elements of the universal set U that are not a member of A. • Denoted A • Example: If A = {x | x is an integer and x < 4} and U = Z, then A = {x | x is an integer and x > 4} • Venn diagram

  8. Complement “With Respect to…” • The complement of B with respect to A – all elements belonging to A, but not to B. • It’s as if U is in the complement is replaced with A. • Denoted A – B = {x | x A and x B} • Example: Assume A = {a, b, c} and B = {b, c, d, e}A – B = {a}B – A = {d, e} • Venn diagram B – A A – B

  9. Symmetric difference • Symmetric difference – If A and B are two sets, the symmetric difference is the set of elements belonging to A or B, but not both A and B. • Denoted AB = {x | (x A and x B) or (x B and x A)} • AB = (A – B)  (B – A) • Venn diagram

  10. Algebraic Properties of Set Operations • Commutative propertiesA  B = B  AA  B = B  A • Associative propertiesA  (B  C) = (A  B)  CA  (B  C) = (A  B)  C • Distributive propertiesA  (B  C) = (A  B)  (A  C)A  (B  C) = (A  B)  (A  C)

  11. More Algebraic Properties of Set Operations • Idempotent propertiesA  A = AA  A = A • Properties of the complement(A) = AA  A = UA  A =  = UU = A  B = A  B -- De Morgan’s lawA  B = A  B -- De Morgan’s law

  12. More Algebraic Properties of Set Operations • Properties of a Universal SetA  U = UA  U = A • Properties of the Empty SetA  = A or A  { } = AA  =  or A  { } = { }

  13. The Addition Principle • The Addition Principle associates the cardinality of sets with the cardinality of their union • If A and B are finite sets, then |A  B| = |A| + |B| – |A  B| • Let’s use a Venn diagram to prove this: • The Roman Numerals indicate how many times each segment is included for the expression |A| + |B| • Therefore, we need to remove one |A  B| since it is counted twice. A  B 1 2 1

  14. Addition Principle Example • Let A = {a, b, c, d, e} and B = {c, e, f, h, k, m} • |A| = 5, |B| = 6, and |A  B| = |{c, e}| = 2 • |A  B| = |{a, b, c, d, e, f, h, k, m}| |A  B| = 9 = 5 + 6 – 2 • If A  B = , i.e., A and B are disjoint sets, then the |A  B| term drops out leaving |A| + |B|

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