Question

Consider the relation \(R=\\{(a, b),(a, c),(c, b),(b, c)\\}\) on the set \(A=\\{a, b, c\\} .\) Is \(R\) reflexive? Symmetric? Transitive? If a property does not hold, say why.

Short answer

The relation \(R\) is neither reflexive nor symmetric, but it is transitive.

Step-by-step solution

Checking Reflexivity

For \(R\) to be reflexive, all elements in set \(A\) must relate to themselves, meaning the pairs \((a,a)\), \((b,b)\) and \((c,c)\) must be in \(R\). Upon checking, these pairs are absent, so \(R\) is not reflexive.

Checking Symmetry

For \(R\) to be symmetric, if any pair \((a,b)\) is in \(R\), then the pair \((b,a)\) must also be in \(R\). The pairs \((a,b)\) and \((b,a)\) are not both present. Similarly, \((a,c)\) and \((c,a)\) are also not both present. Thus, \(R\) is not symmetric.

Checking Transitivity

For \(R\) to be transitive, if \((a,b)\) and \((b,c)\) are in \(R\), then \((a,c)\) must also be in \(R\). Checking the pairs, \((a,b)\), \((b,c)\) and \((a,c)\) are in \(R\), meaning \(R\) is transitive.

Key concepts

Reflexivity

Understanding the concept of reflexivity in relations is fundamental in mathematics. A relation is considered reflexive if every element relates to itself. In other words, for a set A, each element a in A needs to include the pair (a, a) in the relation for it to be reflexive.

For instance, let's take the relation R on the set A ={a, b, c}. To test for reflexivity, we look for the presence of the pairs (a, a), (b, b), and (c, c) in R. If even one of these pairs is missing, the relation cannot be classified as reflexive. In our exercise, since none of these pairs appear in the given relation R, it fails to meet the criteria for reflexivity. This is a helpful tip for quickly identifying non-reflexive relations in homework problems or exams.

Symmetry

The symmetry property in the context of relations requires that for every pair (a, b) in a relation R, the inverse pair (b, a) should also be in R. Symmetry is about balance; whatever is applicable in one direction should be applicable in the opposite direction as well.

In our exercise, the relation R includes pairs like (a, b), but lacks the corresponding (b, a). This absence indicates that R is not symmetric. A good trick to remember is that symmetric relations are like mirrors reflecting pairs; each pair should be flanked by its mirror image within the relation set. Remembering this can save time when you're trying to quickly determine if a relation exhibits symmetry.

Transitivity

Transitivity is another cornerstone in understanding relations. A relation is considered transitive if, whenever it includes pairs (a, b) and (b, c), it also contains the pair (a, c). It is a property that ensures consistency in the sequence of related elements.

Looking at the given relation R from our exercise, we see it has (a, b) and (b, c). Since it also includes (a, c), we can confirm that R is transitive. Recognizing transitivity helps in creating logical links in a sequence of elements and is crucial for making sound mathematical arguments. Monitoring these patterns can greatly enhance problem-solving skills in algebra and beyond.