HelmutKnaust at 21:00, 15 November 2013

2013-11-15T21:00:05Z

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..."

2013-11-05T05:01:34Z

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
\[(m,n)\in R \Leftrightarrow m^2+n^2 \mbox{ is even.}\]
#Show that $R$ is an equivalence relation.
#Find all distinct equivalence classes of this relation.

'''Problem 28.''' Let $R$ and $S$ be two equivalence relations on a non-empty set $X$. Prove or disprove:
#$R\cap S$ is an equivalence relation.
#$R\cup S$ is an equivalence relation.

'''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.

'''Problem 30.''' Consider the following relation $R$ defined on a Boolean Algebra ${\cal A}$:
\[(P,Q)\in R \Leftrightarrow P\sqcup Q=Q\]
Prove or disprove: $R$ is (a) reflexive, (b) transitive, (c) symmetric, (d) anti-symmetric.

@@ Line 4: / Line 4: @@
 '''Problem 27.'''  Let $R$ be a relation on $\mathbb{N}$ defined by
-\[(m,n)\in R \Leftrightarrow m^2+n^2 \mbox{ is even.}\]
+\[(p,q)\in R \Leftrightarrow p^2+q^2 \mbox{ is even.}\]
 #Show that $R$ is an equivalence relation.
 #Find all distinct equivalence classes of this relation.

CRN 12107: HW 6 - Revision history

HelmutKnaust at 21:00, 15 November 2013

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..."