Given that 𝑋 equals the set
containing negative seven, negative one, nine and 𝑅 is the set containing the
ordered pairs 𝑎, negative one; 𝑏, negative seven; negative seven, nine, where 𝑅
is a function on 𝑋, find the numerical value of 𝑎 plus 𝑏.
Let’s begin by looking at some of
the notation and terminology in the question. We’re told that 𝑅 is a function on
𝑋. This means that 𝑅 takes values in
the set 𝑋 and maps them onto exactly one value in a second set. The values in 𝑋 are the input to
the function. So we say the 𝑋 is its domain. Let’s define then the range of the
function to be some second set 𝑌. This is the set of possible outputs
to the function. And in fact, we’re told that 𝑋
contains the elements negative seven, negative one, nine. So, what does 𝑌 contain?
Well, the ordered pairs can help
us. The first value in each ordered
pair is the input. So we choose that from set 𝑋. Then, the second value is the
output. So we choose that from set 𝑌. This means set 𝑌 must contain the
values negative one, negative seven, nine. And in fact, we can identify one of
the mappings straightaway. The third ordered pair tells us
that negative seven is mapped onto nine through function 𝑅. So what does this mean for our
other two ordered pairs?
Consider the first one: 𝑎,
negative one. We know that exactly one number in
set 𝑋 will map onto the number negative one in set 𝑌. Similarly, exactly one number, 𝑏,
in set 𝑋 will map onto the number negative seven. For the sake of this question, it
actually doesn’t matter if we define 𝑎 to be negative one and 𝑏 to be nine or the
other way round. In the first case, if we let 𝑎 be
equal to negative one and 𝑏 be equal to nine, our first two ordered pairs become
negative one, negative one and nine, negative seven.
This is equivalent to the mapping
diagram we’ve drawn. But we could, of course, map nine
onto negative one and negative one onto negative seven. We don’t know which one is which,
but we do know that, in either case, when we add 𝑎 and 𝑏, we are going to be
adding negative one and nine. Remember, addition is commutative,
so this can be done in any order. So negative one plus nine is the
same as nine plus negative one. It’s eight. And so we found a numerical value
of 𝑎 plus 𝑏; it’s eight.