MAT 251 Discrete Mathematics

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MAT 251 Discrete Mathematics

• Sets and Functions

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Section 2.2 Operations on Sets

• Def Let A and B be two sets. The union of the
  sets A and B, written A U B, is the set that
  contains elements that are either in A or in B,
  or in both.
• A U B x x ? A or x ? B

B
A
U
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Section 2.2 Operations on Sets

• Def Let A and B be two sets. The intersection
  of the sets A and B, written A ? B, is the set
  that contains elements that are in both A and B.
• A ? B x x ? A and x ? B

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Section 2.2 Operations on Sets

• Note When the intersection of two sets is the
  empty set, we say that the two sets are disjoint.
• Example Let A a, 2, 3 and B 1, 5.
• Then A and B are disjoint, so A ? B Ø.

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Section 2.2 Operations on Sets

• Example Suppose U N
• Let A 1, 2, 4, 51, 59, 60, 100, 250,
• B 2, 4, , 50 and C 1, 51, 59. Find
• a) B U C
• b) A U C
• c) A ? C
• d) A ? B
• e) B ? C

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Section 2.2 Operations on Sets

• The Principle of Inclusion Exclusion gives
• A U B A B - A ? B

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Section 2.2 Operations on Sets

• Def Let A and B be two sets. The difference of
  A and B, written A - B, is the set that contains
  elements that are in A but not in B. Also, called
  the complement of B wrt A.
• A - B x x ? A and x ? B

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Section 2.2 Operations on Sets

• Def Let U be the universal set. The completment
  of the set A, written Ac is the set that
  contains elements that are in U and not in A.
  The complement of A is the set, U-A.
• Ac x x ? A

U
A
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Section 2.2 Operations on Sets

• Set Identities Page 124
• 1) Identity Law
• 2) Domination Law
• 3) Idempotent Law
• 4) Complementation Law
• 5) Commutative Law
• 6) Associative Law
• 7) Distributive Law
• 8) De Morgans Law
• 9) Absorption Law
• 10) Complement Law

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Section 2.2 Operations on Sets

• How do we prove that two sets A and B are
  equal?
• 1) Show A B iff A ? B AND B ? A.
• OR
• 2) Use the set builder definitions and show
  equivalence that way.

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Section 2.2 Operations on Sets

• Prove that A U B B U A.
• Pf We will show this by showing that
• A U B ? B U A and B U A ? A U B.
• First, suppose x ? A U B. Then by definition x
  ? A or x ? B. Since disjunction is commutative,
  we can say that x ? B or x ? A. So, by the
  definition of union we have that
• x ? B U A. So, indeed A U B ? B U A.
• Next, suppose that x ? B U A. Then by
  definition
• x ? B or x ? A. Since disjunction is
  commutative, we can say that x ? A or x ? B. So,
  by the definition of union we have that
• x ?A U B. So, indeed B U A ? A U B.
• So, since each set is a subset of the other we
  have shown that the two sets are equal to each
  other.

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Section 2.2 Operations on Sets

• Prove that A U B B U A.
• Pf We will show this using the following
  steps.
• A U B x x ? A or x ? B
• x x ? B or x ? A
• B U A.

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Section 2.2 Operations on Sets

• Def Membership table are similar to truth
  tables and are set up in the same way. We can
  demonstrate that an element is in a set by using
  a 1, and to indicate that an element is not in a
  set by using a 0.
• We can prove set identities using membership
  tables as well, very similar to proving logic
  rules.

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Section 2.2 Operations on Sets

• Show that (A ? B ? C)c Ac U Bc U Cc

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Section 2.2 Operations on Sets

• Def The union of a collection of sets is the
  set that contains those elements that are members
  of at least one set in the collection. Notation
• The intersection of a collection of sets is
  the set that contains those elements that are
  members of all the sets in the collection.
  Notation

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Section 2.2 Operations on Sets

• Do Page 131/45

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Section 2.2 Operations on Sets

• The notes have been created with the use of
  Discrete mathematics and Its Applications, Sixth
  Edition by K. H. Rosen