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..."

2017-04-10T03:48:49Z

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.''' 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.''' 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.

'''Problem 30.''' On the set of natural numbers $\mathbb{N}$ consider the partial order \[n\ |\ m \Leftrightarrow\ n\mbox{ is a divisor of }m.\]
Draw a ''Hasse diagram'' for the set $A=\{1,2,3,4,5,\ldots, 12,13,14,15\}$ endowed with this partial order.

21495: HW 6 - Revision history

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..."