Exam 9: A: Relations

Suppose |A| = n. Find the number of binary relations on A.

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

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

find the matrix that represents the given relation. Use elements in the order given to determine rows and columns of the matrix. - $R \text { on } \{ 1,2,4,8,16 \} \text {, where } a R b \text { means } a < b \text {. }$

determine whether the binary relation is: (1) reflexive, (2) symmetric, (3) antisymmetric, (4) transitive. -The relation R on the set of all subsets of {1, 2, 3, 4} where SR T means

determine whether the binary relation is: (1) reflexive, (2) symmetric, (3) antisymmetric, (4) transitive. - $\text { The relation } R \text { on } \mathcal { Z } \text { where } a R b \text { means that the units digit of } a \text { is equal to the units digit of } b \text {. }$

List all the binary relations on the set {0, 1}.

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

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

If R = {(1, 2), (1, 4), (2, 3), (3, 1), (4, 2)}, find the reflexive closure of R.

List the relations on the set {0, 1} that are neither reflexive nor irreflexive.

Suppose the relation R is defined on the set Z where aR b means that ab ≤ 0. Determine whether R is an equivalence relation on Z .