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# nasex1 - NUMBERS AND SETS EXAMPLES SHEET 1 W T G 1 Let A B...

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NUMBERS AND SETS EXAMPLES SHEET 1. W. T. G. 1. Let A , B and C be three sets. Give a proof that A \ ( B C ) = ( A \ B ) ( A \ C ) using the criterion for equality of sets. 2. The symmetric difference A 4 B of A and B is defined to be ( A \ B ) ( B \ A ). (That is, it is the set of elements that belong to one of A and B but not both.) Write out a truth table to show that the operation 4 is associative. Show that x belongs to A 4 ( B 4 C ) if and only if x belongs to an odd number of the sets A , B and C and use this observation to give a second proof that 4 is associative. 3. Let A , B , C and D be sets. Prove that A × ( B C ) = ( A × B ) ( A × C ). Is it necessarily true that ( A × B ) ( C × D ) = ( A C ) × ( B D )? 4. Write down the negations of the following statements. (i) n is even or m is a multiple of 3. (ii) Every x A is also an element of B C . (iii) If it is not raining today then no pigs can fly. 5. Let f and g be functions and let h = g f . If f and g are injections/surjections, prove that h is an injection/surjection.

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nasex1 - NUMBERS AND SETS EXAMPLES SHEET 1 W T G 1 Let A B...

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