Problem 1
Identify the equivalence classes of \(\mathrm{N}\) for the following relations: (a) \(\equiv(\bmod 4)\) (b) \(\equiv(\bmod 6)\)
Problem 1
(a) Draw the diagram to represent the \(\mid\) (divides) partial order on \\{1,2,3,4,5,6\\} (b) List all the maximal, maximum, minimal, and minimum elements.
Problem 1
Which of the following relations on the set of all people are reflexive? Symmetric? Antisymmetric? Transitive? Prove your assertions. (a) \(R(x, y)\) if y makes more money than \(x\). (b) \(R(x, y)\) if \(x\) and \(y\) are about the same height. (c) \(R(x, y)\) if \(x\) and \(y\) have an ancestor in common. (d) \(R(x, y)\) if \(x\) and \(y\) are the same sex. (e) \(R(x, y)\) if \(x\) and \(y\) both collect stamps. (f) \(R(x, y)\) if \(x\) and \(y\) like some of the same music.
Problem 2
(a) Draw a diagram to represent the \(\mid\) (divides) partial order on \(\\{0,1,2,3,4,5,6,7\), \(8,9,10,11 \mid\) (b) Identify all minimal, minimum, maximal, and maximum elements in the diagram.
Problem 2
Let \(M=\\{1,2, \ldots, 10] .\) Define a relation \(R\) on elements \(x, y \in M\) such that \((x, y) \in\) \(R\) if and only if there is a positive integer \(k\) such that \(x=k y\). Find the elements of \(R\).
Problem 2
Determine which of the following five relations defined on \(\mathbb{Z}\) are equivalence relations: (a) \(\\{(a, b) \in \mathbb{Z} \times \mathbb{Z}:(a>0\) and \(b>0)\) or \((a<0\) and \(b<0)\\}\) (b) \(\\{(a, b) \in \mathbb{Z} \times \mathbb{Z}:(a \geq 0\) and \(b>0\) ) or \((a<0\) and \(b \leq 0)\\}\) (c) \(\\{(a, b) \in \mathbb{Z} \times \mathbb{Z}:|a-b| \leq 10\\}\) (d) \(\\{(a, b) \in \mathbb{Z} \times \mathbb{Z}:(a \leq 0\) and \(b \geq 0)\) or \((a \leq 0\) and \(b \leq 0)\\}\) (e) \(\\{(a, b) \in \mathbb{Z} \times \mathbb{Z}:(a \geq 0\) and \(b \geq 0\) ) or \((a \leq 0\) and \(b \leq 0)\\}\)
Problem 3
(a) Draw a diagram to represent the \(\mid\) (divides) partial order on the set \(\\{1,2,3,4,5,6\). 7,8,9,10,11\\} (b) Identify all minimal, minimum, maximal, and maximum elements in the diagram.
Problem 3
Find the elements in each of the following relations defined on \(\mathbb{R}\) :
(a) \((x, y) \in R\) if and only if \(x+1
Problem 3
Which of the following relations on the set of all people are reflexive? Symmetric? Antisymmetric? Transitive? Explain why your assertions are true. (a) \(R(x, y)\) if \(x\) and \(y\) either both like German food or both dislike German food. (b) \(R(x, y)\) if (i) \(x\) and \(y\) either both like Italian food or both dislike it, or (ii) \(x\) and \(y\) either both like Chinese food or both dislike it. (c) \(R(x, y)\) if \(y\) is at least two feet taller than \(x\).
Problem 3
Find the elements in the relation "have the same remainder when divided by \(8^{\prime \prime}\) if the relation is defined on \(\\{1,2,3, \ldots, 24,25\\} .\) Also, find the distinct equivalence classes of this equivalence relation.
