Question

Consider the relation \(R=\\{(0,0),(\sqrt{2}, 0),(0, \sqrt{2}),(\sqrt{2}, \sqrt{2})\\}\) on \(\mathbb{R}\). Is \(R\) reflexive? Symmetric? Transitive? If a property does not hold, say why.

Short answer

The relation R is not reflexive since it doesn't contain all the possible pairs \((a,a)\) for all \(a\) in \(\mathbb{R}\). However, R is symmetric as all pairs \((a,b)\) have a corresponding pair \((b,a)\) in R. Lastly, R is transitive as per the given set, all pairs fulfill the conditions of transitivity.

Step-by-step solution

Checking Reflexivity

To check for reflexivity, look for the pairs \((a,a)\) for all \(a\) in R. Given that our set is within \(\mathbb{R}\), which is the set of all real numbers, R should contain every pair of real numbers \((a, a)\) to be considered reflexive. However, R only contains \((0, 0)\) and \((\sqrt{2}, \sqrt{2})\). Therefore, we can conclude that R is not reflexive given that it doesn't contain all real number pairs.

Checking Symmetry

To check for symmetry, if a pair \((a,b)\) is in R, then \((b,a)\) should also be in R. The pairs in R are \((0,0), (\sqrt{2}, 0),(0, \sqrt{2}),(\sqrt{2}, \sqrt{2})\). Each pair has a corresponding reversed pair within the set so we can conclude that R is symmetric.

Checking Transitivity

To check for Transitivity, if \((a,b)\) and \((b,c)\) are in R, then \((a, c)\) should also be in R. In checking our set, it's confirmed that all pairs obey this rule, thus the relation R is transitive.