M122F07A2Solns - Math 122 F01 and F02 2007 Assignment 2...

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Math 122 F01 and F02 2007 Assignment 2 Solutions 1. Let A and B be sets ( i.e. subsets of some universe U ). (a) [10] Suppose A contains at least two elements. Prove that if every proper subset of A is a subset of B , then A B . (Hint: what does it mean for a subset of A with size one to be a subset of B ?) Since A has at least two elements, for every x A the set { x } is a proper subset of A . By the condition, each of these is a subset of B . From the definitions, { x } ⊆ B x B . Therefore, for every x ∈ U , if x A then x B . That is, A B . (b) [4] Give an example and an explanation to demonstrate that the implication in part (a) can be false if A contains only one element. Let A = { 1 } and B = { 2 } . The only proper subset of A is the empty set, which is a subset of B . Thus, every proper subset of A is a subset of B , but A 6⊆ B . 2. [10] Prove that A = B if and only if P ( A ) = P ( B ) . ( ) Suppose
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