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Unformatted text preview: ⊆ B ∩ C . ] Suppose that x is an element in A ∩ C . By deﬁnition of intersection of sets, it follows that x ∈ A and x ∈ C . But by the assumption that A ⊆ B , since x ∈ A , x ∈ B . Therefore x ∈ B and x ∈ C which implies by deﬁnition of the intersection of sets that x ∈ B ∩ C . Thus, A ∩ C ⊆ B ∩ C . 3. (7.2 #21; 8 points ) Deﬁne Floor : R → Z by the formula Floor( x ) = b x c , for all real numbers x . (a) Is Floor one-to-one? Prove or give a counterexample. (b) Is Floor onto? Prove or give a counterexample. Solution: (a) Floor is NOT one-to-one since Floor(1) = b 1 c = 1 = b 1 . 5 c = Floor(1 . 5) while 1 6 = 1 . 5. (b) Floor : R → Z is onto since if n ∈ Z , Floor( n ) = b n c = n . That is, since the ﬂoor of any integer is the integer itself, we can say that Floor maps R ONTO the set of integers. 2...
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This note was uploaded on 04/14/2008 for the course MATH 15A taught by Professor Briggs during the Fall '04 term at UCSD.
- Fall '04