# What is the reflexive closure of R?

• Nov 2nd 2013, 06:54 AM
lamentofking
What is the reflexive closure of R?
Let R be the relation {(a,b) | a (is not equal to) b} on the set of integers. What is the reflexive closure of R?
• Nov 2nd 2013, 07:06 AM
HallsofIvy
Re: What is the reflexive closure of R?
Do you know what these words mean? A relation is "reflexive" if, for every "a" in the set, it is true that aRa. The "reflexive closure" of a relation is the smallest set of pairs, containing the given relation that is reflexive. Here aRb as long as a is not equal to b. What do you get if you add nRn for every integer n to this set?
• Nov 2nd 2013, 08:24 AM
lamentofking
Re: What is the reflexive closure of R?
Quote:

Originally Posted by HallsofIvy
Do you know what these words mean? A relation is "reflexive" if, for every "a" in the set, it is true that aRa. The "reflexive closure" of a relation is the smallest set of pairs, containing the given relation that is reflexive. Here aRb as long as a is not equal to b. What do you get if you add nRn for every integer n to this set?

Every ordered pair?
• Nov 2nd 2013, 10:19 AM
HallsofIvy
Re: What is the reflexive closure of R?
Yes. A reflexive relation contains all pairs of the form "(a, a)". This relation, as given, contains all other pairs so the "reflexive closure" of this relation is the set of all ordered pairs of integers.