Let a b be sets of binary strings the concatenation

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Let A , B be sets of binary strings. The concatenation product of A and B is AB := { ab : a A , b B } Recall that the cartesian product of two sets is A × B := { ( a , b ) : a A , b B } . Example : Let A = { 0 , 01 , 11 } and B = { 1 , 11 } . Section 1 (C. Sloss) MATH 239: May 26, 2008
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Introduction to Binary Strings Unique generation 5. Definition of Unique Generation Definition : If | A × B | = | AB | , then we say AB is uniquely generated. Example : A = { 0 , 01 , 11 } and B = { 0 , 11 } . If AB is uniquely generated, then Φ AB ( x ) = Φ A ( x B ( x ) . AB is uniquely generated if and only if “delete commas and parentheses” is a bijection from A × B to AB . Section 1 (C. Sloss) MATH 239: May 26, 2008
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Introduction to Binary Strings Unique generation 6. Star of a set Definition : A * ; = { } ∪ A AA AAA ∪ · · · = k 0 A k . For binary strings, A k is the concatenation product unless otherwise indicated. By convention, A 0 = { } . The set of all binary strings is { 0 , 1 } * . Section 1 (C. Sloss) MATH 239: May 26, 2008
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Introduction to Binary Strings Unique generation 7. Generating series for A * Definition : A * is uniquely generated if | A * | = { } ∪ A A × A A × A × A ∪ · · · .
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