### Video Transcript

Given that 𝑥 is the set of numbers
20, one, and three and 𝑅 is a relation on 𝑥, where 𝑎𝑅𝑏 means that 𝑎 plus two
𝑏 is equal to an even number for each 𝑎 that exists in 𝑥 and 𝑏 that exists in
𝑥, determine the relation 𝑅.

We are told in the question that
set 𝑥 contains three numbers: 20, one, and three. Our values of 𝑎 and 𝑏 are
contained in set 𝑥. Therefore, they must each be one of
these three numbers. We are also told that 𝑅 is a
relation on 𝑥. Any relation is a set of ordered
pairs 𝑥, 𝑦. Each of our pairs must satisfy that
𝑎 plus two 𝑏 is an even number, where 𝑎 is our 𝑥-value and 𝑏 is our
𝑦-value. We could write down the nine
possible ordered pairs as shown: one, one; one, three; one, 20; three, one; and so
on. We could then substitute each of
these into the expression 𝑎 plus two 𝑏 to identify which are even and which are
odd.

This could be quite time-consuming,
so we will look for a quicker method based on the equation in this question. Multiplying any integer by two
gives an even number. This means that two 𝑏 will always
be even. We recall that adding an even
number to another even number gives an even answer, whereas adding an odd number and
an even number gives an odd answer. In order for 𝑎 plus two 𝑏 to be
even, then 𝑎 must be even. The only one of the three values in
set 𝑥 that is even is 20. This means that 𝑎 must be equal to
20. 𝑏 on the other hand could take any
one of the three values as multiplying any of them by two would give an even
answer. The relation 𝑅 is therefore the
set of three ordered pairs: 20, one; 20, three; and 20, 20.