𝑋 is equal to the set of numbers four, eight, 12, and 𝑅 is a given relation on 𝑋, where 𝑎 𝑅𝑏 signifies that a number 𝑎 is a multiple of 𝑏. Given that 𝑎 is an element of 𝑋 and 𝑏 is an element of 𝑋, write the relation 𝑅.
We recall that a relation is a collection of ordered pairs 𝑎, 𝑏 that satisfy a given rule. In this question, our value 𝑎 must be a multiple of 𝑏. In this question, both 𝑎 and 𝑏 are contained in the set of numbers four, eight, and 12. The multiples of four are the numbers in the four times table: four, eight, 12, 16, and so on. The multiples of eight are eight, 16, 24, and so on. And the multiples of 12 are 12, 24, 36, and so on.
We want the first number in our ordered pair to be a multiple of the second number. This means that we have the ordered pairs four, four; eight, four; and 12, four. We can also have the ordered pairs eight, eight and 12, 12 as eight is a multiple of eight and 12 is a multiple of 12. The relation 𝑅 consists of the set of five ordered pairs: four, four; eight, four; 12, four; eight, eight; and 12, 12.