17.5: Activities
In each of the following, describe the requested combination of relations in words (i.e. in the form “a is related to b if …”). Try to “simplify” your description, if possible.
In Task h and Task i, the symbol \(\equiv_k\) represents a relation on \(\mathbb{Z}\text{,}\) where \(m \equiv_k n\) means that \(m\) and \(n\) have the same remainder when divided by \(k\text{.}\) (It may help to know that this is equivalent to \(k\) dividing the difference \(m - n\text{.}\))
- \(< \cup >\) on \(\mathbb{Z}\text{.}\)
- Union of “longer than” and “shorter than” on \(\Sigma^\ast\) for some alphabet \(\Sigma\text{.}\)
- Union of “longer than”, “shorter than”, and “same length as” on \(\Sigma^\ast\) for some alphabet \(\Sigma\text{.}\)
- Intersection of “longer than” and “shorter than” on \(\Sigma^\ast\) for some alphabet \(\Sigma\text{.}\)
- The complement of \(\le\) on \(\mathbb{Z}\text{.}\)
- The inverse of \(\le\) on \(\mathbb{Z}\text{.}\)
- The inverse of “\(x R y\) if \(2 x + 3 y = 0\)” on \(\mathbb{Z}\text{.}\)
- The intersection of \(\equiv_5\) and \(\equiv_7\) on \(\mathbb{Z}\text{.}\)
- The intersection of \(\equiv_2\) and \(\equiv_4\) on \(\mathbb{Z}\text{.}\)
In each of the following, you are given a set \(A\) and a relation \(R\) on \(A\text{.}\) Determine which of the properties reflexive , symmetric , antisymmetric , and transitive \(R\) possesses.
- \(A = \mathbb{Z}\text{,}\) \(R\) is \(<\text{.}\)
- \(A\) is the set of all straight lines in the plane, \(R\) means “is parallel to.”
- \(A\) is the set of all straight lines in the plane, \(R\) means “is perpendicular to.”
- \(A = \Sigma^\ast\) for some alphabet \(\Sigma\text{,}\) \(R\) means “is the same length as.”
- \(A = \Sigma^\ast\) for some alphabet \(\Sigma\text{,}\) \(R\) means “is shorter than.”
- \(A = \Sigma^\ast\) for some alphabet \(\Sigma\text{,}\) \(x\) is some fixed choice of letter in \(\Sigma\text{,}\) \(R\) means “contains the same number of occurrences of \(x\) as.”
- \(A\) is an arbitrary set, \(R\) is the empty relation .
- \(A\) is an arbitrary set, \(R\) is the universal relation .
- Suppose \(R\) is a relation on a set \(A\text{.}\) Convince yourself that \(R \cup R^{-1}\) is symmetric. (See the Symmetric Relation Test .)
- Recall that \(\vert\) represents the relation “divides” on sets of integers. Draw the directed graph for \(\vert\) on the set \(A = \{2,4,6,8,10,12,14,16\}\text{.}\) Then describe how to obtain the graph for the symmetric relation \(\vert \cup \vert ^{-1}\) as an undirected graph from the graph of \(R\) using only an eraser.
For each of the properties reflexive , symmetric , antisymmetric , and transitive , carry out the following.
Assume that \(R\) and \(S\) are nonempty relations on a set \(A\) that both have the property. For each of \(R^C\text{,}\) \(R \cup S\text{,}\) \(R \cap S\text{,}\) and \(R^{-1}\text{,}\) determine whether the new relation
- must also have that property;
- might have that property, but might not; or
- cannot have that property.
Any time you answer Statement i or Statement iii, outline a proof. Any time you answer Statement ii, provide two examples: one where the new relation has the property, and one where the new relation does not. (You may use graphs to describe your examples.)