Exam 9: Relations

What is the covering relation of the partial ordering $\{ ( a , b ) \mid a \text { divides } b \} \text { on the set } \{ 2,4,6,8,10,12 \} ?$

In questions give an example or else prove that there are none. - $\text { A relation on } \{ a , b , c \} \text { that is reflexive and transitive, but not antisymmetric. }$

If $\mathbf { M } _ { R } = \left( \begin{array} { l l l l } 1 & 1 & 1 & 1 \\0 & 1 & 1 & 1 \\0 & 0 & 1 & 1 \\0 & 0 & 0 & 1\end{array} \right)$ determine if R is: (a) reflexive (b) symmetric (c) antisymmetric (d) transitive.

In questions determine whether the binary relation is: (1) reflexive, (2) symmetric, (3) antisymmetric, (4) transitive. -The relation R on {a, b, c} where R = {(a, a), (b, b), (c, c), (a, b), (a, c), (c, b)}.

List the reflexive relations on the set {0, 1}.

In questions find the matrix that represents the given relation. Use elements in the order given to determine rows and columns of the matrix. - $R \text { on } \{ 1,2,3,4,6,12 \} \text { where } a R b \text { means } a \mid b \text {. }$

In questions determine whether the binary relation is: (1) reflexive, (2) symmetric, (3) antisymmetric, (4) transitive. - $\text { The relation } R \text { on the set } \{ ( a , b ) \mid a , b \in \mathcal { Z } \} \text { where } ( a , b ) R ( c , d ) \text { means } a = c \text { or } b = d \text {. }$

Let R be the relation on A={1,2,3,4,5} where R={(1,1),(1,3),(1,4),(2,2),(3,1),(3,3),(3,4),(4,1) , (4,3),(4,4),(5,5)} . R is an equivalence relation. Find the equivalence classes for the partition of A given by R .

supp $R = \{ ( a , b ) , ( a , d ) , ( b , c ) , ( c , c ) , ( d , a ) \} \text { and } S = \{ ( a , c ) , ( b , d ) , ( d , a ) \}$ ose R and S are relations on {a, b, c, d}, where Find the combination of relations. - $S ^ { 2 }$

Suppose the relation R is defined on the set $\mathcal { Z }$ where aRb means that $a b < 0$ Determine whether R is an equivalence relation on $\mathcal { Z }$

Suppwhetherose RR isis thean equivrelationalenceonrelationN whereonaRNb. means that a ends in the same digit in which b ends. Determine

If R= $\{ ( x , y ) \mid x$ and $y$ are bit strings containing the same number of 0s}, find the equivalence classes of (a) 1 . (b) 00 . (c) 101 .

supp $R = \{ ( a , b ) , ( a , d ) , ( b , c ) , ( c , c ) , ( d , a ) \} \text { and } S = \{ ( a , c ) , ( b , d ) , ( d , a ) \}$ ose R and S are relations on {a, b, c, d}, where Find the combination of relations. - $R \circ S$

Suppose A is the set composed of all ordered pairs of positive integers. Let R be the relation defined on A where (a, b) R(c, d) means that a+d=b+c . (a) Prove that R is an equivalence relation. (b) Find [(2,4)] .

What is the covering relation of the partial ordering $\{ ( a , b ) \mid a \text { divides } b \} \text { on the set } \{ 1,2,3,4,6,8,12,24 \} ?$

In questions determine whether the binary relation is: (1) reflexive, (2) symmetric, (3) antisymmetric, (4) transitive. - $\text { The relation } R \text { on } \mathcal { R } \text { where } a R b \text { means } a - b \in \mathcal { Z } \text {. }$

Suppose A={2,3,6,9,10,12,14,18,20} and R is the partial order relation defined on A where xRy means x is a divisor of y . (a) Draw the Hasse diagram for R . (b) Find all maximal elements. (c) Find all minimal elements. (d) Find lub({2,9}) . (e) Find lub ({3,10}) . (f) Find glb ({14,10}) .

In questions find the matrix that represents the given relation. Use elements in the order given to determine rows and columns of the matrix. - $R \text { on } \{ w , x , y , z \} \text { where } R = \{ ( w , w ) , ( w , x ) , ( x , w ) , ( x , x ) , ( x , z ) , ( y , y ) , ( z , y ) , ( z , z ) \} \text {. }$

Find the transitive closure of R on {a, b, c, d} where R={(a, a),(b, a),(b, c),(c, a),(c, c),(c, d),(d, a),(d, c)} .

In questions determine whether the binary relation is: (1) reflexive, (2) symmetric, (3) antisymmetric, (4) transitive. -The relation R on Z where aR b means $a \neq b$