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Unformatted text preview: ELEN 303: Assignment 1 Instructor: Dr. Jean-Fran¸ cois Chamberland Email: [email protected] (Subject ECEN 303) Office: Room 244F WERC Office Hours: Tue 2:30 - 3:45 p.m. Problems: 1. Following the argument presented in the notes, prove that parenleftBigg intersectiondisplay α ∈ A S α parenrightBigg c = uniondisplay α ∈ A S c α . Suppose that x belongs to (intersectiontext α ∈ I S α ) c . That is, there exists an α ′ ∈ I such that x is not an element of S α ′ . This implies that x belongs to S c α ′ , and therefore x ∈ uniontext α ∈ I S c α . Thus, we have shown that (intersectiontext α ∈ I S α ) c ⊂ uniontext α ∈ I S c α . The converse is obtained by reversing the argument. Suppose that x belongs to uniontext α ∈ I S c α . Then, there exists an α ′ ∈ I such that x ∈ S c α ′ . This implies that x / ∈ S α ′ and, as such, x / ∈ (intersectiontext α ∈ I S α ) . Alternatively, we have x ∈ (intersectiontext α ∈ I S α ) c ....
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