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# sols6 - 14 6 September 15th Naive set theory basic...

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14 6. September 15th: Naive set theory: basic operations. Relations. 6.1 . The useful abbreviation x A, p ( x ) stands for ‘the statement p ( x ) is true for all x in the set A ’. In other words, ‘for all x , if x is an element of A , then p ( x ) is true’. More formally, it is shorthand for x ( x A )= p ( x ) . What is the formal way to interpret the abbreviation x A, p ( x )? Answer: x A, p ( x ) should be interpreted as x ( x A ) p ( x ) . This should make sense without much further discussion: there exists an x in the set A such that p ( x ) is true really means there exists an x such that x is in A and p ( x ) is true. If this is not formal enough for you, remember that x, a ( x ) is equivalent to ¬ ( x, ¬ a ( x ) (this was discussed in § 3.2). By the same token, x A, p ( x ) is equivalent to ¬ ( x A, ¬ p ( x )), or, formally, (*) ¬ ( x, ( x A ⇒ ¬ p ( x )) . Now recall that a = b is equivalent to ( ¬ a ) b ; therefore, (*) is equivalent to ¬ ( x, ( x ±∈ A ) ∨ ¬ p ( x )) therefore to x, ¬ (( x ±∈ A ) p ( x )) and Fnally to x, ( x A ) p ( x ) by one of De Morgan’s laws. ± 6.2 . Prove parts (7) and (8) of Theorem 6.1. Answer: These statements are S ± ( A B ) = ( S ± A ) ( S ± B ) S ± ( A B ) = ( S ± A ) ( S ± B ) . They are set-theoretic incarnations of De Morgan’s laws. ±or instance, consider the second formula; let p, q be the statements x A , x B respectively, and let x S . Then x S ± ( A B ) means x ±∈ A B , that is, ¬ ( x A B ), that is, ¬ (( x A ) ( x B )), by deFnition of union of sets. This is ¬ ( p q ) .

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sols6 - 14 6 September 15th Naive set theory basic...

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