23666: HW 6 - Revision history

HelmutKnaust at 22:26, 11 April 2019

2019-04-11T22:26:37Z

HelmutKnaust at 22:24, 11 April 2019

2019-04-11T22:24:16Z

HelmutKnaust: Created page with "'''Problem 26.''' Consider the following equivalence relation on the set $A=\{1,2,3,4,5,6\}$: \[R=\{(1,1),(2,2),(3,3),(4,4),(5,5),(6,6),(1,2),(1,4),(2,1),(2,4),(4,1),(4,2),(3..."

2019-04-11T22:22:32Z

Created page with "'''Problem 26.''' Consider the following equivalence relation on the set $A=\{1,2,3,4,5,6\}$: \[R=\{(1,1),(2,2),(3,3),(4,4),(5,5),(6,6),(1,2),(1,4),(2,1),(2,4),(4,1),(4,2),(3..."

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'''Problem 26.''' Consider the following equivalence relation on the set $A=\{1,2,3,4,5,6\}$:
\[R=\{(1,1),(2,2),(3,3),(4,4),(5,5),(6,6),(1,2),(1,4),(2,1),(2,4),(4,1),(4,2),(3,6),(6,3)\}.\]
Find the partition generated by $R$.

'''Problem 27.''' Let $R$ be a relation on $\mathbb{N}$ defined by
\[(p,q)\in R \Leftrightarrow p^2+q^2 \mbox{ is even.}\]
#Show that $R$ is an equivalence relation.
#Find all distinct equivalence classes of this relation.

'''Problem 28.''' Define a relation $R$ on $\mathbb{N}$ by: $xRy$ if x and y have the same prime divisors. (For eaxmple, $6R12$.)
#Show that $R$ is an equivalence relation.
#Find the partition generated by $R$.

'''Problem 29.''' A relation $R$ on a non-empty set $X$ is called ''reverse-transitive'' if
\[(a,b)\in R \wedge (b,c)\in R \Rightarrow (c,a)\in R \mbox{ for all } a,b,c \in X.\]
Show that a relation $R$ on a non-empty set $X$ is an equivalence relation if and only if it is reflexive and reverse-transitive.

@@ Line 13: / Line 13: @@
 '''Problem 29.'''  A relation $R$ on a non-empty set $X$ is called ''reverse-transitive'' if
-\[(a,b)\in R \wedge (b,c)\in R \Rightarrow (c,a)\in R$ for all $a,b,c \in X.\]
+\[(a,b)\in R \wedge (b,c)\in R \Rightarrow (c,a)\in R \mbox{ for all }a,b,c \in X.\]
 Show that a relation $R$ on a non-empty set $X$ is an equivalence relation if and only if it is reflexive and reverse-transitive.

@@ Line 8: / Line 8: @@
 #Find all distinct equivalence classes of this relation.
-'''Problem 28.''' Define a relation $R$ on $\mathbb{N}$ by: $xRy$ if x and y have the same prime divisors. (For eaxmple, $6R12$.)
+'''Problem 28.''' Define a relation $R$ on $\mathbb{N}$ by: $xRy$ if $x$ and $y$ have the same prime divisors. (For example, $6R12$.)
 #Show that $R$ is an equivalence relation.
 #Find the partition generated by $R$.
 '''Problem 29.'''  A relation $R$ on a non-empty set $X$ is called ''reverse-transitive'' if
-\[(a,b)\in R \wedge (b,c)\in R \Rightarrow (c,a)\in R \mbox{ for all } a,b,c \in X.\]
+\[(a,b)\in R \wedge (b,c)\in R \Rightarrow (c,a)\in R$ for all $a,b,c \in X.\]
 Show that a relation $R$ on a non-empty set $X$ is an equivalence relation if and only if it is reflexive and reverse-transitive.