Let R be the relation on the set of all colorings of the 2 ×2 checkerboard where each of the fou

Let R be the relation on the set of all colorings of the 2...

Let $R$ be the relation on the set of all colorings of the $2 \times 2$ checkerboard where each of the four squares is colored either red or blue so that $\left(C_{1}, C_{2}\right),$ where $C_{1}$ and $C_{2}$ are $2 \times 2$ checkerboards with each of their four squares colored blue or red, belongs to $R$ if and only if $C_{2}$ can be obtained from $C_{1}$ either by rotating the checkerboard or by rotating it and then reflecting it. a) Show that $R$ is an equivalence relation. b) What are the equivalence classes of $R ?$

Let $R$ be the relation on the set of all colorings of the $2 \times 2$ checkerboard where each of the four squares is colored either red or blue so that $\left(C_{1}, C_{2}\right),$ where $C_{1}$ and $C_{2}$ are $2 \times 2$ checkerboards with each of their four squares colored blue or red, belongs to $R$ if and only if $C_{2}$ can be obtained from $C_{1}$ either by rotating the checkerboard or by rotating it and then reflecting it.
a) Show that $R$ is an equivalence relation.
b) What are the equivalence classes of $R ?$

Key Concepts

Symmetry Transformations

Symmetry transformations include operations like rotations and reflections that map an object onto itself in ways that preserve its inherent structure. These transformations are key in determining when two configurations are considered the same, as they capture the inherent symmetrical properties of the object in question.

Group Actions on Sets

Group actions describe the way a group, composed of elements such as rotations and reflections, can systematically interact with and transform the elements of a set. This concept is central in understanding how symmetries can partition a set into classes where elements are equivalent if one can be transformed into another via the group's actions.

Equivalence Relation

An equivalence relation is a binary relation that is reflexive, symmetric, and transitive. It allows a set to be partitioned into distinct equivalence classes where each element of the set is equivalent to every other element in its class, based on the defining properties of the relation.

Equivalence Classes

Equivalence classes are subsets formed by grouping together all elements that are equivalent under a given equivalence relation. Each element of the original set belongs to one and only one equivalence class, and these classes form a partition of the set.

Transcript

00:03 We are given a relation on the set of all colorings of the 2x2 checkerboard.

00:10 This relation says that, well, we have that the 2 by 2 checkerboards, c1 and c2, if each of their four squares, colored blue or red, belongs to this relation, if and only if c2 can be obtained from c1, either by rotating the checkerboard or by retaining it and then reflecting.

00:37 In part a, where i should show this relation is, in fact, a equivalence relation.

00:43 So first, let's prove the reflexivity.

00:55 So, let's be a coloring of a two -by -two checkerboard, where squares are red or blue? well, clearly, it follows that checkerboard b can be obtained by rotating checkerboard b 360 degrees.

02:15 So it follows that b is related to b, and therefore, we have that the relation r is reflexive.

02:30 Now let's put symmetry.

02:32 So suppose that b1 is related to b2.

02:43 Now this means that b2 can be obtained from b1 by rotating and or reflecting b1.

03:15 Now, if we rotate by x degrees, then we can rotate x degrees in the other direction to obtain b1 from b2.

03:59 Now, if you rotate by x degrees and reflect, then we can reflect and rotate and rotate in the opposite direction by x degrees to obtain b1 from b2.

04:52 So it follows that b1 can also be obtained from b2 that rotating into reflecting b2.

05:19 And therefore it follows that b2 is also related to b1, and so it follows that r is symmetric.

05:30 Finally, we want to prove that r is transitive, so suppose that b1 is related to b2 and that b2 is related to b3.

05:50 This means that b2 can be obtained from b1 by rotating and or reflecting, and likewise b3 can be obtained by rotating or reflecting b2, so, if we rotate by x degrees to obtain v2 from b1, and rotate by y degrees to obtain b3 from b2, and we can obtain b3 from b1, or retaining x plus y degrees, and performing the same reflections required to obtain b2 from b2, and to obtain b2 from b1.

08:14 So we've shown that b3 can be obtained from b1, and therefore it follows that b1 is related to b3.

08:22 And therefore, we have that r is transitive.

08:28 Since r is reflexive, symmetric, and transitive, r is an equivalence relation.

08:46 In part b, we're asked to find the equivalence classes of r.

08:54 For the purpose of this problem, we have that there are two colors each square to take on.

09:01 B for blue, or r for red.

09:19 Now, let x, y, z, b, the equivalent, classes of r are going to be, well, we have the equivalence class of the board with upper left b, upper right blue, lower left blue, lower left blue, lower right blue.

09:45 Well, this is simply going to be the board b, b, b, b, b, b itself...

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