Video: Writing a Relation on a Set given Its Rule

Given that 𝑋 = {20, 1, 3}, and 𝑅 is a relation on 𝑋, where 𝑎𝑅𝑏 means that 𝑎 + 2𝑏 is equal to an even number for each 𝑎 ∈ 𝑋 and 𝑏 ∈ 𝑋, determine the relation 𝑅.

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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.