Week 7 Sets

Week 7 Sets - CONCORDIA UNIVERSITY DEPARTMENT OF COMPUTER...

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CONCORDIA UNIVERSITY COMP 232/2 Mathematics for Computer Science FALL 2010 Sets Set notation is an alternate notation for predicate calculus. Often a statement is clearer in set notation than in the standard notation of predicate calculus. Deﬁnitions. The set builder notation for a propositional function P ( x ) is y ∈ { x | P ( x ) } def P ( y ) . The construction { x | P ( x ) } is called a set . Some speciﬁc sets are N def = { x | x is a natural number } , Z def = { x | x is an integer } , Q def = { x | x is a rational number } , R def = { x | x is a real number } . Variations of set builder notation are often used: { x 1 , ..., x n } denotes { x | ( x = x 1 ) ∨ ··· ∨ ( x = x n ) } ; { x S | P ( x ) } denotes { x | x S P ( x ) } ; { F ( x ) | P ( x ) } denotes { x | ∃ z ( P ( z ) x = F ( z ) ) } . So we can write N = { x Z | x 0 } , Q = { n/d | n Z , d Z , d 6 = 0 } . Examples. y ∈ { x 2 | x Z } ⇐⇒ y = 0 , 1 , 4 , 9 , 16 , 25 , .... y ∈ { x R | x 2 Z } ⇐⇒ y = 0 , ± 1 , ± 2 , ± 3 , ± 2 , ± 5 , .... n ∈ { 2 h 3 k | h Z , k Z , (0 < h < k < 5) } ⇐⇒ ∃ h k ( h Z k Z (0 < h < k < 5) ( n = 2 h 3 k ) ) ⇐⇒ n ∈ { 18 , 54 , 108 , 162 , 324 , 648 } . Deﬁnitions. negation x / A def ≡ ¬ ( x A ) subset A B def ≡ ∀ x ( x A x B ) equality A = B def ≡ ∀ x ( x A x B

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This note was uploaded on 11/15/2010 for the course ENCS COMP 232 taught by Professor Ford during the Fall '10 term at Concordia CA.

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Week 7 Sets - CONCORDIA UNIVERSITY DEPARTMENT OF COMPUTER...

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