8 Let A 1234 and Bab How many relations are there betwen the

8. Let A ={1,2,3,4} and B={a,b}

How many relations are there betwen the set A and B? (Do not give just a number but explain how you would compute it.

Solution

Consider the set A={1, 2, 3, 4} (There’s nothing special about this

ticular set; any set of numbers would do for this example.) Elements of
can be compared to each other by the symbol “<.” For example, 1 < 4,
, 2 < 4, and so on. You have no trouble understanding this because the
tion of numeric order is so ingrained. one with an obsession for detail but absolutely no
understanding of the meaning of (or relationships between) integers. You
might consider writing down for your student the following set:
R={(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)}

The set R encodes the meaning of the < relation for elements in A. An
ordered pair B= (a, b) appears in the set if and only if a < b. If asked whether
or not it is true that 3 < 4, through R until the ordered pair (3, 4); then know 3 < 4 is true. If asked
about 4<2, . The set
R, which is a subset of A*B, completely describes the relation < for A.
Though it may seem simple-minded at rst, this is exactly the idea
we will use for our main denition. This denition is general enough to
describe not just the relation <for the set A={1, 2, 3, 4}, but any relation

for any set A.