We have a relation R on Z+ defined as follows: mRn if and only if m|n.
a. Explain why the relation R is not a function.
b. Determine the set A = {m ∈ Z|mR52} and give its cardinality |A|.
c. Determine the set B = {n ∈ Z|52Rn}.
d. Indicate whether A ∩ B = ∅.
(a) clearly 2R4 and 2R6. for a function, the each m must be related to a single n
(b) A is the set of factors of 52. I'm sure you can list them.
(c) B is the set of all multiples of 52.
(d) A ∩ B = {52}
a. To determine whether the relation R is a function, we need to check if each element in the domain has a unique mapping to the target set. In this case, the domain is Z+ (the set of positive integers) and the target set is also Z+.
For a relation to be a function, each element in the domain should match to at most one element in the target set. However, in the given relation R, an element m can have multiple elements n in the target set where m | n. For example, 2R4 and 2R6, so 2 has multiple mappings to different elements in the target set. Hence, the relation R is not a function.
b. To determine the set A = {m ∈ Z | mR52}, we need to find all the positive integers m such that m divides 52.
The positive divisors of 52 are: 1, 2, 4, 13, 26, and 52. So, the set A contains these divisors: A = {1, 2, 4, 13, 26, 52}.
The cardinality of set A (|A|) represents the number of elements in set A, which in this case is 6.
c. To determine the set B = {n ∈ Z | 52Rn}, we need to find all the positive integers n such that 52 divides n.
The positive multiples of 52 are: 52, 104, 156, 208, and so on. So, the set B contains these multiples: B = {52, 104, 156, 208, ...}.
Note that B contains infinitely many elements since there are infinitely many multiples of 52.
d. To determine whether A ∩ B = ∅ (the intersection of sets A and B is empty), we need to check if there are any common elements between sets A and B.
In this case, A = {1, 2, 4, 13, 26, 52} and B = {52, 104, 156, 208, ...}.
The only common element between A and B is 52. So, A ∩ B is not empty since the intersection contains at least one element, which is 52.