a) Define an equivalence relation. b) Which relations on the set {a, b, c, d} are equivalence relations and contain (a, b) and (b, d)?

Let R and S be equivalence relations on a set A. Prove that R=S if and only if the equivalence classes of R are the same as the equivalence classes of S.

a. Is

$$3 \in \{ 1,2,3 \} ?$$

, b. Is

$$1 \subseteq \{ 1 \} ?$$

, c. Is

$$\{ 2 \} \in \{ 1,2 \} ?$$

, d. Is

$$\{ 3 \} \in \{ 1 , \{ 2 \} , \{ 3 \} \} ?$$

, e. Is

$$1 \in \{ 1 \} ?$$

, f. Is

$$\{ 2 \} \subseteq \{ 1 , \{ 2 \} , \{ 3 \} \} ?$$

, .g Is

$$\{ 1 \} \subseteq \{ 1,2 \} ?$$

, h. Is

$$1 \in \{ \{ 1 \} , 2 \} ?$$

, i. Is

$$\{ 1 \} \subseteq \{ 1 , \{ 2 \} \} ?$$

, j. Is

$$\{ 1 \} \subseteq \{ 1 \} ?$$

Determine which of the reflexive, symmetric, or transitive properties hold for these relations. Which are equivalence relations?

$$\{(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)\}$$

Can someone explain equivalence classes in discrete math?

Let X = {0, 1}. (a) List (as subsets of $X \times X)$ all possible relations on X. (b) Which of the relations in part (a) are equivalence relations?

Define three equivalence relations on the set of students in your discrete mathematics class different from the relations discussed in the text. Determine the equivalence classes for each of these equivalence relations.

Explain the relationship between equivalence relations on a set and partitions of this set.

Determine whether the relation defined on the set of positive integers is reflexive, symmetric, antisymmetric, transitive, and/or a partial order. (x,y)$\in$R if 3 divides x+y

Let R be a reflexive relation on a set A. Show that

$$R^n$$

is reflexive for all positive integers n.

Determine which of the reflexive, symmetric, and transitive properties are satisfied by the given relation $R$ defined on set $S$.

$S$ is the set of positive integers and $x R y$ means that $x$ divides $y$ or $y$ divides $x$.

For each of the following relations defined on the set {1, 2, 3, 4, 5}, determine whether the relation is reflexive, irreflexive, symmetric, antisymmetric, and/or transitive. a. R = {(1,1),(2,2),(3,3),(4,4),(5,5)} b. R = {(1,2),(2,3),(3,4),(4,5)} c. R = {(1,1),(1,2),(1,3),(1,4),(1,5)} d. R = {(1,1),(1,2),(2,1),(3,4),(4,3)} e. R = {1,2,3,4,5}$\times${1,2,3,4,5}

Define an equivalence relation on the set $\mathbf{R}$ that partitions the real line into subsets of length $1$.

Prove that refines is a partial order relation on the set of all partitions of a set A.