Exam 9: A: Relations

find the matrix that represents the given relation. Use elements in the order given to determine rows and columns of the matrix. - $R \text { on } \{ - 2 , - 1,0,1,2 , \} , \text { where } a R b \text { means } a ^ { 2 } = b ^ { 2 } \text {. }$

If X=( Fran Williams, 617885197, MTH 202, 248B West ) , find the projections $P _ { 1,3 } ( X ) \text { and } P _ { 1,2,4 } ( X )$

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

give an example or else prove that there are none. -A relation on {1, 2, 3} that is reflexive and transitive, but not symmetric.

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

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

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

determine whether the binary relation is: (1) reflexive, (2) symmetric, (3) antisymmetric, (4) transitive. -The relation R on {w, x, y, z} where R = {(w, w), (w, x), (x, w), (x, x), (x, z), (y, y), (z, y), (z, z)}.

give an example or else prove that there are none. -A relation on {1, 2} that is symmetric and transitive, but not reflexive.

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

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

determine whether the binary relation is: (1) reflexive, (2) symmetric, (3) antisymmetric, (4) transitive. -The relation R on the set of all people where aR b means that a is younger than b.

Draw the Hasse diagram for the relation $R \text { on } A = \{ 2,3,4,6,10,12,16 \} \text { where } a R b \text { means } a \mid b$

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)}. Draw the directed graph for R.

Find the smallest equivalence relation on {1, 2, 3} that contains (1, 2) and (2, 3).

List the relations on the set {0, 1} that are reflexive and symmetric.

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. - $S ^ { 2 }$

Find the smallest partial order relation on {1, 2, 3} that contains (1, 1), (3, 2), (1, 3).

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