ps06 - MAT 300 Mathematical Structures Dr L Mantini Problem...

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Unformatted text preview: MAT 300, Mathematical Structures Dr. L. Mantini Problem Set 6 Solutions Spring 2010 4.1 : 3, 4, 5, 6, 7, 10, 12; 6.1 : 2, 5, 6, 9, 12 4.1.3 Describe the truth sets of the following statements as subsets of R 2 (pictures omitted). (a) y = x 2- x- 2: region is graph of the parabola opening up with x-intercepts at (2 , 0) and (- 1 , 0) (b) y < x : region is the open half-plane below the line y = x (c) either y = x 2- x- 2 or y = 3 x- 2: region is union of given parabola and line (d) y < x , and either y = x 2- x- 2 or y = 3 x- 2: region is the portion of the graph from (c) that lies strictly below the line y = x 4.1.5 Prove parts 2 and 3 of Theorem 4.1.3. Solution : We prove part 2 as a two-part proof showing both set containments sepa- rately, and we prove part 3 using a string of equivalent statements. 2. Prove that A × ( B ∪ C ) = ( A × B ) ∪ ( A × C ). Proof : ( ⊆ ) Let ( x,y ) ∈ A × ( B ∪ C ). Then x ∈ A and y ∈ B ∪ C . So either y ∈ B or y ∈ C . If y ∈ B , then ( x,y ) ∈ A × B . If y ∈ C , then ( x,y ) ∈ A × C . In either case, ( x,y ) ∈ ( A × B ) ∪ ( A × C ). ( ⊇ ) Now let ( x,y ) ∈ ( A × B ) ∪ ( A × C ). So either ( x,y ) ∈ A × B or ( x,y ) ∈ A × C . If ( x,y ) ∈ A × B , then x ∈ A and y ∈ B ⊆ B ∪ C , so ( x,y ) ∈ A × ( B ∪ C ). If ( x,y ) ∈ A × C , then x ∈ A and y ∈ C ⊆ B ∪ C , so ( x,y ) ∈ A × ( B ∪ C ). In either case, ( x,y ) ∈ A × ( B ∪ C ), which was what we had to prove. 3. Prove that ( A × B ) ∩ ( C × D ) = ( A ∩ C ) × ( B ∩ D ). Proof : Let x and y be arbitrary. Then ( x,y ) ∈ ( A × B ) ∩ ( C × D ) ⇔ ( x,y ) ∈ A × B ∧ ( x,y ) ∈ C × D ⇔ x ∈ A ∧ y ∈ B ∧ x ∈ C ∧ y ∈ D ⇔ ( x ∈ A ∧ x ∈ C ) ∧ ( y ∈ B ∧ y ∈ D ) ⇔ x ∈ A ∩ C ∧ y ∈ B ∩ D ⇔ ( x,y ) ∈ ( A ∩ C ) × ( B ∩ D ) ....
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ps06 - MAT 300 Mathematical Structures Dr L Mantini Problem...

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