We will explore relations that are antisymmetric and asymmetric in both a real-world context and a mathematical context. We will examine properties of each of these types of relations, and determine how best to tell them apart by their differences.
Suppose that Riverview Elementary is having a father son picnic, where the fathers and sons sign a guest book when they arrive.
Here’s something interesting! This list of fathers and sons and how they are related on the guest list is actually mathematical! In mathematics, a relation is a set of ordered pairs, (x, y), such that x is from a set X, and y is from a set Y, where x is related to y by some property or rule.
If we let F be the set of all fathers at the picnic, and S be the set of all sons, then the guest book list, call it G, is a relation from set F to set S. That is, G consists of all the ordered pairs (f, s), such that f is related to s by the rule that f is the father of s.Let’s consider another example of a relation in the real world that wouldn’t seem mathematical at first glance. Consider the relation A that is defined by the rule ‘is a relative that came before that individual (an ancestor), or is that individual’. In other words, A is the set of ordered pairs (a, b), such that a is a relative of b that came before b, or is b. Once again, one wouldn’t think a list of pairs such as this would be mathematical, but it is!
Asymmetric and Antisymmetric Relations
When it comes to relations, there are different types of relations based on specific properties that a relation may satisfy.
Two of those types of relations are asymmetric relations and antisymmetric relations. Ot the two relations that we’ve introduced so far, one is asymmetric and one is antisymmetric. Let’s take a look at each of these types of relations and see if we can figure out which one is which.An asymmetric relation, call it R, satisfies the following property:
- If (x, y) is in R, then (y, x) is not in R.
Therefore, if an element x is related to an element y by some rule, then y cannot be related to x by that same rule. In other words, in an asymmetric relation, it can’t go both ways.
An antisymmetric relation, call it T, satisfies the following property:
- If (x, y) and (y, x) are in T, then x = y.
That is, if an element x is related to an element y, and the element y is also related to the element x, then x and y must be the same element. Thus, in an antisymmetric relation, the only way it can go both ways is if x = y.
Okay, similar names, but we can see that an asymmetric relation is different from an antisymmetric relation in that an asymmetric relation absolutely cannot go both ways, and an antisymmetric relation can go both ways, but only if the two elements are equal.
Let’s think about our two real-world examples of relations again, and try to determine which one is asymmetric and which one is antisymmetric. First, consider the relation G consisting of ordered pairs (f, s), such that f is the father of s. Hmmm…for this relation to be asymmetric, it would have to be the case that if (f, s) is in G, then (s, f) cannot be in G. This makes sense! If f is the father of s, then s certainly can’t be the father of f. That would be biologically impossible! Therefore, G is asymmetric, so we know it is not antisymmetric, because the relation absolutely cannot go both ways.Now, consider the relation A that consists of ordered pairs, (a, b), such that a is the relative of b that came before b or a is b.
In order for this relation to be antisymmetric, it has to be the case that if (a, b) and (b, a) are in A, then a = b. Again, this makes sense! If a is a relative of b that came before b or is b and b is a relative of a that came before a or is a, then it must be the case that a and b are the same person, because it can’t be the case that a came before b and b came before a. Therefore, the only possibility is that a is b. Since it is possible for it to go both ways in this relation (as long as a = b), the relation is antisymmetric, but can’t be asymmetric.If you’re wondering about some examples that actually seem more mathematical.
Consider the relations < and ≤, where (a, b) is in < only if a is strictly less than b, and (c, d) is in ≤ only if c is less than or equal to d. The relation < is asymmetric, because it can’t be the case that for two numbers, a and b, a ; b and b ; a, so if (a, b) is in ;, then (b, a) can’t be in <. It absolutely can’t go both ways.On the other hand, the relation ; is antisymmetric, because if for two numbers c and d, both c ; d and d ; c, then it must be the case that c = d.
The only way for it to go both ways is if c = d.
A relation is a set of ordered pairs, (x, y), such that x is related to y by some property or rule. Two types of relations are asymmetric relations and antisymmetric relations, which are defined as follows:
- Asymmetric: If (a, b) is in R, then (b, a) cannot be in R.
- Antisymmetric: If (a, b) and (b, a) are in R, then a = b.
The easiest way to remember the difference between asymmetric and antisymmetric relations is that an asymmetric relation absolutely cannot go both ways, and an antisymmetric relation can go both ways, but only if the two elements are equal.As we’ve seen, relations (both asymmetric and antisymmetric) can easily show up in the world around us, even in places we wouldn’t expect, so it is great to be familiar with them and their properties!