# p4 - b A A B c A ∩ B A#4 The symmetric diﬀerence of A...

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Math 5615H: Introduction to Analysis I. Fall 2011 Homework #4 (due on Wednesday, October 5). 50 points are divided between 5 problems, 10 points each. #1. Let f be a mapping of A to B . Show that for each B 1 B and B 2 B , their inverse images satisfy the properties ( i ) f - 1 ( B 1 B 2 ) = f - 1 ( B 1 ) f - 1 ( B 2 ) , ( ii ) f - 1 ( B 1 B 2 ) = f - 1 ( B 1 ) f - 1 ( B 2 ) . #2. Let f be a mapping of A to B . Verify whether of not the images of subsets A 1 A and A 2 A in general satisfy the properties ( i ) f ( A 1 A 2 ) = f ( A 1 ) f ( A 2 ) , ( ii ) f ( A 1 A 2 ) = f ( A 1 ) f ( A 2 ) . #3 The set diﬀerence of A and B by deﬁnition is A \ B := { x A : x / B } . Simplify the expressions ( a ) A \ ( B \ A )
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Unformatted text preview: , ( b ) A \ ( A \ B ) , ( c ) A ∩ ( B \ A ) . #4. The symmetric diﬀerence of A and B by deﬁnition is A ∆ B := ( A \ B ) ∪ ( B \ A ). Show that for an arbitrary set C , A ∆ B ⊂ ( A ∆ C ) ∪ ( B ∆ C ) . #5. Show that the unit interval I 1 := { x : 0 ≤ x ≤ 1 } is equivalent to the unit square I 2 := { x = ( x 1 , x 2 ) : 0 ≤ x 1 , x 2 ≤ 1 } . Hint. Use decimal representation of x ∈ I 1 ....
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