Rec0524_key - COT 3100 Summer 2001 Recitation 05/24 Sets 1 Let U is all real numbers and consider the following sets which are intervals of real

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Unformatted text preview: COT 3100 Summer 2001 Recitation 05/24 Sets 1. Let U is all real numbers and consider the following sets, which are intervals of real numbers: A =[1,3)={ x | 1 ≤ x <3 } B = (2, 4]={ x | 2< x ≤ 4 } C = (-∝ , 5) ={ x | x < 5) Express the following sets as intervals, or unions of disjoint intervals: a) ¬ C ¬ C = { x | x ≥ 5 } b) A - B A - B = { x |1 ≤ x ≤ 2 } c) A - C A - C= ∅ d) C - A C - A= { x | x < 1 or 3 ≤ x <5 } e) A ∩ B A ∩ B = { x | 1< x <3 } f) A ∪ B A ∪ B = { x | 1 ≤ x ≤ 4 } 2. Prove or disprove that for any sets A , B , C a) A ∪ B - A ∩ B =( A - B ) ∪ ( B - C ) A ∪ B - A ∩ B = { x | x ∈ A ∪ B ∧ x ∉ A ∩ B }..………by definition of the set difference = { x | x ∈ A ∪ B ∧ ¬ ( x ∈ A ∩ B )}..……………………by definition of ∉ = { x | ( x ∈ A ∨ x ∈ B ) ∧ ¬ ( x ∈ A ∧ x ∈ B )}..………by definition of ∪ and ∩ = { x | ( x ∈ A ∨ x ∈ B ) ∧ ( x ∉ A ∨ x ∉ B )}..………………by DeMorgan’s law = { x | [ x ∈ A ∧ ( x ∉ A ∨ x ∉ B )] ∨ [ x ∈ B ∧ ( x ∉ A ∨ x ∉ B )]}.… distributive law)]}....
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Rec0524_key - COT 3100 Summer 2001 Recitation 05/24 Sets 1 Let U is all real numbers and consider the following sets which are intervals of real

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