Which of these collections of subsets are partitions of the set of integers? a)

Question

Which of these collections of subsets are partitions of the set of integers?

a) the set of eeven integers and the set of odd integers
b) the set of positive integers and the set of negative integers
c) the set of integers divisible by 3, the set of integers leaving a remainder of 1 when divided by 3, and the set of integers leaving a remainder of 2 when divided by 3
d) the set of integers less than - 100, the set of integers with absolute value not exceeding 100, and the set of integers greater than 100
e) the set of integers not divisible by 3, the set of even integers, and the set of integers that leave a remainder of 3 when divided by 6


When you answer the question, please add exaplaination
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Explanation / Answer

a) is a partition because it covers all integers.

b) is NOT a partition, because it doesn;t cover zero.

c) is a partition because it covers all integers.

d) is a partition because it covers all integers.

e) is a partition because it covers all integers. an integer is either (not divisible by 3) or is of the form 6k or 6k+3, 6k is even and 6k+3 leave a remainder of 3 when divided by 6.

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