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m325kprac3a

# m325kprac3a - 7 Give a precise deﬁnition of...

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M325K Prac. Test 3 4.3–5.3 1. Let U = { 1 , 2 , 3 , . . . , 10 } be the universal set and let A = { 2 , 4 , 6 } and B = { 1 , 4 , 5 } . Find the number of elements in A c B c , A c Δ B c , and P ( A × B ). 2. Use induction to prove that the sum of the first n odd positive integers equals n 2 . 3. Prove that 2 n + 1 < n 3 for all integers n 2. 4. Prove that if A B , then A B = B . 5. Use mathematical induction to prove that for all positive integers n , 8 | 3 2 n - 1. 6. How many mistakes are in the sentence “Ther are three mistakes in thiss sen- tence”? Justify your answer.
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Unformatted text preview: 7. Give a precise deﬁnition of “partition” of a set. 8. State the Well-Ordering Principle for the integers. 9. Use mathematical induction to show that 6 | n ( n 2 + 5) for all positive integers n . 10. Let A,B, and C be sets. Us an element argument to prove that A ∪ ( B ∩ C ) = ( A ∪ B ) ∩ ( A ∪ C ) . 11. Let A and B be sets. Construct an algebraic proof that A-( A ∩ B ) = ( A-B ) ....
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