Given that 𝑋 is equal to the set
zero, negative one and 𝑌 is equal to the set eight, negative four, negative three,
negative two, then which of the following relations would negative one, negative
four be an element of? Is it option (A) 𝑋 squared, (B) 𝑌
squared, (C) 𝑋 times 𝑌, or (D) 𝑌 times 𝑋?
The times notation in options (C)
and (D) corresponds to the Cartesian product of sets 𝑋 and 𝑌 and sets 𝑌 and 𝑋,
respectively. The Cartesian product is the set of
all ordered pairs. Before starting this question, it
is worth noting that option (A) 𝑋 squared is the same as the Cartesian product of
set 𝑋 and set 𝑋. Likewise, 𝑌 squared is equal to
the Cartesian product of set 𝑌 and set 𝑌.
The first value of our ordered
pair, negative one, occurs in set 𝑋. The second value, negative four,
occurs in set 𝑌. We can therefore conclude that
option (C) the Cartesian product of 𝑋 and 𝑌 is the correct answer. This relation contains the ordered
pair negative one, negative four.
An alternative method here would be
to list all the ordered pairs of each relation. For example, the Cartesian product
of 𝑋 and 𝑌 contains the ordered pairs zero, eight; zero, negative four; zero,
negative three; zero, negative two; negative one, eight; negative one, negative
four; negative one, negative three; and negative one, negative two. Once again, we see that negative
one, negative four is contained in this Cartesian product.