I am studying computer science and I have Maths exam soon. While solving past exam papers, I encountered with a sets question and couldn't solve. It consists of two parts (a) and (b) and couldn't solve part b but I am going to include all parts as it is required in order to solve the question.
Question is as follows:
(a) Let D be the set of decimal digits, D = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, and let E and L be the following subsets of D:
E = { n ∈ D : n/2 ∈ D} , L = {n ∈ D : n < 3}
Express the following sets as explicit lists of members, treating D is the Universal Set:
(i) E ⋃ L
(ii) E' ⋂ L
(iii) E x L
(b) The members of the Cartesian product D x D can be paired with members of the set C = {0,1, ... ,99} in the following way:
(i,j) ∈ D x D pairs with n = 10i + j ∈ C
Subsets of D x D can be paired with subsets of C in the same way; for example
L x D ⊂ D x D pairs with {n ∈ C : n < 30} ⊂ C
Find expressions, in terms of D, E and L, for the subsets of D x D that pair with the following subsets of C:
(i) A = {0,2,4,6, ...,96,98}
(ii) B = {0,1,2,20,21,22}
As I said, I only couldn't solve part (b) so I only need answers for part (b).
Many thanks for all of the answers in advance.
B)(i): Expressing $\{0,2,4,6,8,10,\dots,96,98\}$ using the available sets.
The idea here is that the unnamed pairing function will take the first entry of the tuple and use that as the tens-digit and the second entry of the tuple and use that as the ones-digit. Notice here that the tens-digit in what we want to express can range over all possible digits while the ones-digit ranges only over the even digits.
$$D\times E = \{(0,0),(0,2),(0,4),\dots,(9,6),(9,8)\}$$
which via the pairing function alluded to mapping $(a,b)\mapsto 10a+b$ gives the desired result
B)(ii): Expressing $\{0,1,2,20,21,22\}$
The idea here is that the tens digit must be even and must be small while the ones-digit simply needs to be small. We can ensure even-ness using $E$. We can ensure small-ness using $L$.
$$(E\cap L)\times L = \{(0,0),(0,1),(0,2),(2,0),(2,1),(2,2)\}$$
which via the pairing function alluded to before mapping $(a,b)\mapsto 10a+b$ gives the desired result.