Binary Operations

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Binary Operations
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Definition

• A binary operation on a nonempty set A is a
  mapping defined on A?A to A, denoted by f A?A ?
  A.

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Ex1. (a)

• Let be the addition operation on Z.
• Z?Z ? Z defined by (a, b) ab
• Let ? be the multiplication on R.
• ? R?R ? R defined by ?(a, b) ab

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Ex1. (b)

• ?Z?Z ? Z defined by ?(x, y) xy?1
• ?(1, 1) ?(2, 3)
• Then ? is a binary operation on Z.
• ?Z?Z ? Z defined by ?(x, y) 1xy
• ?(1, 1)
• ?(2, ?3)
• Then ? is a binary operation on Z.

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Ex1. (c)

• Let be the division operation on Z.
• Then (1, 2)½. (1, 2)?Z?Z , but ½?Z.
• Thus is not a binary operation.
• If we deal with on R , then is not a
  binary operation, either.
• Because (a , 0) is undefined.
• But is a binary operation on R?0.

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Ex2.

• The intersection and union of two sets are both
  binary operations on the universal set .

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Definitions

• If ? is a binary operation on the nonempty set
  A, then we say ? is commutative if
• x ? y y ? x, ?x, y?A.
• If x ? (y ? z) (x ? y) ? z, ? x, y, z ? A,
• then we say that the binary operation is
  associative.

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Ex3.(a)

• The Operations and ? on Z are both
  commutative and associative.

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Ex3. (b)

• But operation Z?Z?Z defined by
• (a, b) a b is not commutative.
• Since
• The operation is not associative, either.
  Because

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Ex4. (a)

• Let ? be the operation defined as Ex1(b) on Z,
  x ? y xy?1. Then ? is both commutative and
  associative.
• Pf

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Ex4. (b)

• Let ? be the operation defined as Ex1(b) on Z,
  x?y 1xy. Then ? is commutative but not
  associative.
• Pf

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Definition

• Let ? A?A ? A is a binary operation on a
  nonempty set A and let B ? A.
• If x?y?B, ?x, y ?B, then we say B is closed with
  respect to ?.

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Ex5.

• (a) The set S of all odd integers is closed with
  respect to multiplication.
• (b) Define ?Z?Z ? Z by x ? y ?x? ?y?.
• Let B be the set of all negative integers. Then
  B is not closed with respect to ?,

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Definition

• Let A be a nonempty set and
• let ? A?A ? A be a binary operation on A. An
  element e ?A is called an (two side) identity
  element with respect to ?
• if e?x x x?e, ?x?A.

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Ex6.

• (a) The integer 1 is an identity w. r. t. ?,
  but not w. r. t. .
• The number 0 is an identity w. r. t. .
• (b) Let ? be the operation defined as Ex1(b) on
  Z, x ? y xy ?1. Then

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Ex6. (continuous)

• (c) Let ? be the operation defined as Ex1(b) on
  Z, x?y 1xy. Then the operation has no
  identity element in Z.
• Pf

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Definition

• Let e be the identity element for the binary
  operation ? on A and a? A.
• If ?b? A such that a?b e (or b?a e)
• then b is called a right inverse
• (or left inverse) of a w. r. t. ?.
• If both a ?b e b ?a, then b (denoted by a?1)
  is called an (two-side) inverse of a
• a?1 is called an invertible element of a.

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Note

• The identity e and the two-side inverse of an
  element w. r. t. a binary operation ? are unique.
• Pf

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Ex7.

• Let ? be the operation defined as Ex1(b) on Z,
  x ? y xy ?1. Then (2x) is a two-side inverse
  of x w. r. t. ?, ?x?Z.
• Pf

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Ex8. (a)

• Give a binary operation on Z as follow.
• (a) x ? y x

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Ex8. (b)

• (b) x ? y x2y. This operation is neither
• associative, nor commutative.
• Pf

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Ex8. (b) (continuous)

• (b) x ? y x 2y.This operation has no
  identity, thus no inverse.
• Pf

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Ex8. (c)

• (c) x ? y x xy y.