Given that 𝑎 is an element of 𝑋 and 𝑏 is an element of 𝑌, express the following arrow diagram in the form of an equation.
In this question, we have an arrow diagram that represents a relation between two sets. This relation takes elements from set 𝑋 and maps them onto elements in set 𝑌. And in fact, we notice that each element in the first set is mapped onto exactly one element in the second set. So, in fact, we’re able to say that this relation represents a one-to-one mapping, and therefore it’s a function.
So, what is the equation we can use to describe this function? Let’s inspect the elements in set 𝑋 and the elements they map onto in set 𝑌. The first mapping takes the number five from set 𝑋 and maps it onto the number three in set 𝑌. We can write this as an ordered pair five, three. The next element eight is mapped onto the element six in set 𝑌. And so, as an ordered pair, that’s eight, six. Finally, element 10 in set 𝑋 is mapped onto element eight in set 𝑌. So as an ordered pair, we have 10, eight.
Now, whilst writing these as ordered pairs isn’t entirely necessary, it can make the next step a little easier to spot. In general, since 𝑎 is an element of set 𝑋 and 𝑏 is an element of set 𝑌, the ordered pairs take the form 𝑎, 𝑏. We now need to identify what the relationship is between 𝑎 and 𝑏 in each pair. Well, we might notice that if we take the first element 𝑎 and subtract two from it, we get the second element 𝑏. Five minus two is three, eight minus two is six, and so on. So, in general, we can say that 𝑎 minus two must be equal to 𝑏. And in fact, this is the equation. This is the equation that represents the arrow or mapping diagram. It’s 𝑏 equals 𝑎 minus two.