mt1sol - Homework 1, Math 245, Fall 2010 Due Wednesday,...

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Homework 1, Math 245, Fall 2010 Due Wednesday, September 15 in class. The solution of each exercise should be at most one page long. If you can, try to write your solutions in LaTex. Each question is worth 2 points. 1. Let A,B,C be sets. Prove that A ( B \ C ) = ( A B ) \ ( A C ). Proof. We prove this statement by double inclusion. First, we show that A ( B \ C ) ( A B ) \ ( A C ). Let x A ( B \ C ). This means x A and x B \ C . The last statement implies x B and x / C . Now, as x A and x B , it follows that x A B . Also, as x A and x / C , we deduce that x / A C (otherwise, x A C would imply x C which is a contradiction with x / C ). Thus, we obtained that x A B and x / A C . Hence, x ( A B ) \ ( A C ) which shows A ( B \ C ) ( A B ) \ ( A C ). Now we prove the opposite inclusion ( A B ) \ ( A C ) A ( B \ C ). Let y ( A B ) \ ( A C ). This means y A B and y / A C . The fact that x A B means that x A and x B . Now as y A and y / A C , we deduce that y / C (otherwise, y C combined with y A would imply y A C which is a contradiction with y / A C ). Thus, we have proved that y A,y B and y / C . The last two facts imply y B \ C . Combining this with y A shows that y A ( B \ C ). Hence, we have proved that ( A B ) \ ( A C ) A ( B \ C ). This finishes the proof of our exercise. 2. Let
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mt1sol - Homework 1, Math 245, Fall 2010 Due Wednesday,...

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