Let A = {1,2,3,4} and B = {3,4,5,6}. Find B × A. Choose the correct complete set of ordered pairs.

Let A = {1,2,3,4}, B = {3,4,5,6}, and C = {2,4,6}. Find A × (B ∪ C). Choose the correct complete set of ordered pairs.

Let A = {1,2,3,4}, B = {3,4,5,6}, and C = {2,4,6}. Find (A ∩ B) × C. Choose the correct complete set of ordered pairs.

With A = {1,2,3,4}, B = {3,4,5,6}, and C = {2,4,6}, evaluate (A × B) ∩ (B × C). Choose the correct set.

With A = {1,2,3,4}, B = {3,4,5,6}, and C = {2,4,6}, evaluate (A × B) − (B × C). Choose the correct set.

For any non-empty sets A, B, and C, which of the following product-set identities are valid? Select all that apply.

For any non-empty sets A, B, and C, which statements are true? Select all that apply.

Suppose A, B, C ⊂ Z × Z with A = {(x,y) | y = 5x − 1}, B = {(x,y) | y = 6x}, and C = {(x,y) | 3x − y = −7}. Find A ∩ B.

Suppose A, B, C ⊂ Z × Z with A = {(x,y) | y = 5x − 1}, B = {(x,y) | y = 6x}, and C = {(x,y) | 3x − y = −7}. Find B ∩ C.

Let the universal set be Z × Z and A, B, C ⊂ Z × Z with A = {(x,y) | y = 5x − 1}, B = {(x,y) | y = 6x}, and C = {(x,y) | 3x − y = −7}. Evaluate A̅ ∪ C̅.

Let the universal set be Z × Z and A, B, C ⊂ Z × Z with A = {(x,y) | y = 5x − 1}, B = {(x,y) | y = 6x}, and C = {(x,y) | 3x − y = −7}. Evaluate B̅ ∪ C̅.

Let A = {1, 2, 3, 4, 6} and R be the relation on A defined by a R b if and only if a is a multiple of b. Choose the correct set of ordered pairs for R.

Let A and B be finite sets with |B| = 3. If there are 4096 relations from A to B, what is |A|?

Let A = {1, 2, 3} and B = {2, 4, 5}. Determine |A × B|.

Let A = {1, 2, 3} and B = {2, 4, 5}. Determine the number of relations from A to B.

Let A = {1, 2, 3}. Determine the number of binary relations on A.

Let A = {1, 2, 3} and B = {2, 4, 5}. Determine the number of relations from A to B that contain both (1, 2) and (1, 5).

Let A = {1, 2, 3} and B = {2, 4, 5}. Determine the number of relations from A to B that contain exactly 5 ordered pairs.

Let A = {1, 2, 3}. Determine the number of binary relations on A that contain at least 7 ordered pairs.

Let A = {1, 2, 3, 4} and R be the relation on A defined by xRy iff y = 2x. Which set of ordered pairs represents R as a subset of A × A?

Let A = {1, 2, 3, 4} and R be the relation on A defined by xRy iff y = 2x. What are the in-degree and out-degree of vertex 2 in the digraph of R?

Let A = {1, 2, 3, 4} and R be the relation on A defined by xRy iff y = 2x. Which adjacency matrix M(R), with rows and columns ordered as 1, 2, 3, 4, is correct?

Let A = {1, 2, 3, 4, 6} and R be the relation on A defined by aRb iff a is a multiple of b. Which set of ordered pairs correctly represents R as a subset of A × A?

For the relation on A = {1, 2, 3, 4, 6} defined by aRb iff a is a multiple of b, what are the in-degree and out-degree of vertex 6 in the digraph of R?

For the relation on A = {1, 2, 3, 4, 6} defined by aRb iff a is a multiple of b, which adjacency matrix M(R) with rows/columns ordered 1, 2, 3, 4, 6 is correct?

For A = {a, b, c, d, e, f}, the digraph shown represents a relation on A. Which ordered pair is included in the relation?

For A = {a, b, c, d, e, f}, based on the digraph shown for the relation on A, which entry of the adjacency matrix M(R) (rows/columns ordered a, b, c, d, e, f) is 1?

Let A = {1, 2, 3, 4} and R be a relation on A defined by (a, b) ∈ R iff a divides b. Which set of ordered pairs represents R as a subset of A × A?

For the relation on A = {1, 2, 3, 4} defined by a divides b, which statement about powers of the relation is correct?

Let A = {1,2,3,4}, B = {w,x,y,z}, and C = {5,6,7}. Relations are defined by R1 from A to B and R2, R3 from B to C as R1 = {(1,x),(2,x),(3,y),(3,z)}, R2 = {(x,6)}, and R3 = {(w,5),(w,6)}. Determine R1 ∘ R2.

Using A = {1,2,3,4}, B = {w,x,y,z}, C = {5,6,7}, with R1 = {(1,x),(2,x),(3,y),(3,z)} and R3 = {(w,5),(w,6)} defined on B × C, find R1 ∘ R3.

With A = {1,2,3,4}, B = {w,x,y,z}, C = {5,6,7}, R1 = {(1,x),(2,x),(3,y),(3,z)}, and R2 = {(x,6)}, which matrix equals M(R1 ∘ R2) when rows are ordered 1,2,3,4 and columns are ordered 5,6,7?

Based on R1 = {(1,x),(2,x),(3,y),(3,z)} and R2 = {(x,6)} over sets A = {1,2,3,4}, B = {w,x,y,z}, and C = {5,6,7}, does M(R1 ∘ R2) equal the Boolean matrix product M(R1) · M(R2)?

Let A be {1, 2, 3, 4} and R be a relation on A defined by R = {(1,2), (1,3), (2,4), (3,2), (3,3), (3,4)}. Which set equals R composed with R, written $$R^2$$ ?

With A = $${1, 2, 3, 4}$$ and R = $${(1,2), (1,3), (2,4), (3,2), (3,3), (3,4)}$$ , which set equals $$R^3$$ (that is, $$R^2$$ composed with $$R$$ )?

For A = {1, 2, 3, 4} and R = {(1,2), (1,3), (2,4), (3,2), (3,3), (3,4)}, which 4×4 adjacency matrix M(R) (rows and columns ordered 1,2,3,4) is correct? Give entries row-wise.

Which 4×4 adjacency matrix M(R^2) (rows and columns ordered 1,2,3,4) matches the digraph of R^2? Give entries row-wise.

Select all statements that are correct for Boolean matrix multiplication with M(R) corresponding to the relation R on A = {1, 2, 3, 4}.

Given the digraphs of relations R and S on the set $$A=\{a,b,c\}$$ as described: R has edges $$\{(a,a),(a,b),(a,c),(b,c)\}$$ and S has edges $$\{(a,c),(b,a),(b,c),(c,c)\}$$ . Which set of ordered pairs is $$R\cup S$$ ?

Using the same digraphs of R on $$A=\{a,b,c\}$$ with edges $$\{(a,a),(a,b),(a,c),(b,c)\}$$ , which set of ordered pairs is the complement $$\overline{R}$$ taken over $$A\times A$$ ?

From the digraphs of $$R$$ and $$S$$ on $$A=\{a,b,c\}$$ where $$R=\{(a,a),(a,b),(a,c),(b,c)\}$$ and $$S=\{(a,c),(b,a),(b,c),(c,c)\}$$ , which set of ordered pairs is $$R\cap S$$ ?

For relation $$R=\{(a,a),(a,b),(a,c),(b,c)\}$$ on $$A=\{a,b,c\}$$ , which set of ordered pairs is the converse (inverse) relation $$R^{c}$$ ?

Let $$A = \{1,2,3\}$$ . For the relation $$R_1 = \{(1,2),(2,1),(1,3),(3,1)\}$$ on $$A$$ , select all properties that hold.

Let $$A = \{1,2,3\}$$ . For the relation $$R_2 = \{(1,1),(2,2),(3,3),(2,3)\}$$ on $$A$$ , select all properties that hold.

Let $$A = \{1,2,3\}$$ . For the relation $$R_3 = \{(1,1),(2,2),(3,3)\}$$ on $$A$$ , select all properties that hold.

Let $$A = \{1,2,3\}$$ . For the relation $$R_4 = \{(1,1),(2,2),(3,3),(2,3),(3,2)\}$$ on $$A$$ , select all properties that hold.

Let $$A = \{1,2,3\}$$ . For the relation $$R_5 = \{(1,1),(2,3),(3,3)\}$$ on $$A$$ , select all properties that hold.

For the relation $$R_6 = \{(2,3),(3,4),(2,4)\}$$ , select all properties that hold.

Let $$A = \{1,2,3\}$$ . For the relation $$R_7 = \{(1,3),(3,2)\}$$ on $$A$$ , select all properties that hold.

Let $$A = \{1,2,3,4\}$$ . Choose a relation on $$A$$ that is reflexive and symmetric but not transitive.

Let $$A = \{1,2,3,4\}$$ . Choose a relation on $$A$$ that is reflexive and transitive but not symmetric.

Let $$A = \{1,2,3,4\}$$ . Choose a relation on $$A$$ that is symmetric and transitive but not reflexive.

Let $$A=\{1,2,3,4\}$$ and $$R=\{(1,1),(1,2),(2,1),(2,2),(3,3),(3,4),(4,3),(4,4)\}$$ be a relation on $$A$$ . Select all properties that $$R$$ satisfies.

Let $$A=\{1,2,3,4,5,6,7,8,9,10,11,12\}$$ . Define a relation $$R$$ on $$A$$ by $$(x,y)\in R$$ if and only if $$x-y$$ is a multiple of $$5$$ . Select all properties that $$R$$ satisfies.

Let $$A=A_1\cup A_2\cup A_3$$ , where $$A_1=\{1,2\}$$ , $$A_2=\{2,3,4\}$$ and $$A_3=\{5\}$$ . Define a relation $$R$$ on $$A$$ by $$xRy$$ if and only if $$x$$ and $$y$$ are in the same set $$A_i$$ for $$i\in\{1,2,3\}$$ . Select all properties that $$R$$ satisfies.

For a fixed integer $$n>1$$ , let $$R$$ be the relation on the set of all integers $$\mathbb{Z}$$ defined by $$aRb$$ if and only if $$a\equiv b \pmod n$$ . Select all properties that $$R$$ satisfies.

For the equivalence relation R = {(1,1), (1,2), (2,1), (2,2), (3,4), (4,3), (3,3), (4,4)} defined on the set $$A=\{1,2,3,4\}$$ , determine the partition of A induced by R.

Find the partition of A induced by R, given $$A=\{1,2,3,4,5,6,7,8,9,10,11,12\}$$ . The relation R is defined by $$(x,y)\in R$$ iff $$x-y$$ is a multiple of $$5$$ .

Let $$A=\{1,2,3,4,5,6,7\}$$ and let R be the equivalence relation on A that induces the partition $$\{\{1,2\},\{3\},\{4,5,7\},\{6\}\}$$ . Identify R from the options below.

Let $$A=\{1,2,3,4,5\}$$ . A relation on $$A\times A$$ is defined by: $$(x_1,y_1)$$ is related to $$(x_2,y_2)$$ if and only if $$x_1+y_1=x_2+y_2$$ . Select all properties that this relation satisfies on $$A\times A$$ .

Let $$A=\{1,2,3,4,5\}$$ . The relation on $$A\times A$$ is: $$(x_1,y_1)$$ is related to $$(x_2,y_2)$$ iff $$x_1+y_1=x_2+y_2$$ . Which set is the equivalence class of $$(1,3)$$ ?

Let $$A=\{1,2,3,4,5\}$$ . The relation on $$A\times A$$ is: $$(x_1,y_1)$$ is related to $$(x_2,y_2)$$ iff $$x_1+y_1=x_2+y_2$$ . Which set is the equivalence class of $$(2,4)$$ ?

Let $$A=\{1,2,3,4,5\}$$ . The relation on $$A\times A$$ is: $$(x_1,y_1)$$ is related to $$(x_2,y_2)$$ iff $$x_1+y_1=x_2+y_2$$ . Which set is the equivalence class of $$(1,1)$$ ?

Let $$A=\{1,2,3,4,5\}$$ and a relation on $$A\times A$$ defined by equality of sums: $$(x_1,y_1)$$ is related to $$(x_2,y_2)$$ iff $$x_1+y_1=x_2+y_2$$ . Which description correctly gives the partition of $$A\times A$$ induced by this relation?

Find the number of equivalence relations that can be defined on a finite set A with $$|A|$$ $$=$$ $$6$$ .

Let A = {1,2,3,4} and R = {(1,1),(1,2),(2,2),(2,4),(1,3),(3,3),(3,4),(1,4),(4,4)} on A. Based on this relation, select all properties that R satisfies on A.

For A = {1,2,3,4} with R as in the previous question, choose the description that matches the Hasse diagram of (A,R).

Let R be the relation on A = {1,2,3,4} defined by xRy iff x divides y. Select all properties that R satisfies on A.

In the Hasse diagram for the divisibility relation on A = {1,2,3,4}, which pair is not a cover relation (i.e., there is an intermediate element between them)?

Let A = {1,2,3,4,6,8,12} and define R by xRy iff x divides y. Select all properties that R satisfies on A.

In the Hasse diagram for the divisibility relation on A = {1,2,3,4,6,8,12}, which elements cover 4? Select all that apply.

Consider the poset of positive divisors of 36 under divisibility, D36 = {1,2,3,4,6,9,12,18,36}. In its Hasse diagram, which elements cover 6? Select all that apply.

In the Hasse diagram of the poset of positive divisors of 36 under divisibility, how many maximal elements are there?