Let T be a square with each side having length 1 located in

Let T be a square with each side having length 1 located in the plane so that the top side is parallel to the x-axis. Let S be the set of coloured squares obtainable from T by painting each side with one of the colours red and blue. Any combination of colours is allowed, for example all sides could have the same colour. Note that S has 16 elements: for example the top edge being red and all others blue is a different painting that the bottom edge being red and all others blue. Define a relation R on S by s_1 R s_2 if and only if s_1 can be rotated in place so that the rotated coloured square is identical to s_2. Prove that R is an equivalence relation and find the 6 equivalence classes. (Your sets can contain pictures of the coloured squares they contain, or you can use abbreviations like RRBR to denote the side colours going clockwise from the top.) Define a relation F on S by s_1 F s_2 if and only if s_1 can be rotated or flipped over a vertical, horizontal or diagonal line so that the resulting coloured square is identical to s_2. Taking for granted that this is an equivalence relation, find the partition of S induced by the equivalence classes.

Solution

a) A relation is said to be reflexive if (a,a) belongs to R

s1 R s1 -> Since s1 can be rotated in order to obtain again s1

hence the relation is reflexive in nature

A relation is said to be symmetric if (a,b) belongs to R, then (b,a) belongs to R

if s1 R s2=> implies s1 can be moved to form s2

Then s2 R s1= > it will also be true, since we can also move s2 to form s1

A relation is said to be transitive if (a,b) and (b,c) belongs to R, then (a,c) belongs to R

s1 R s2 => s1 can be rotated to form s2

s2 R s3 => s2 can be rotated to form s3

s1 R s3 => (Rotate s1 to obtain s2) + (Now rotate modified s2 to obtain s3) which is possible from the second

Hence s1 R s3 is possible

Therefore, relation is reflexive,symmetric and transitive

A relation is said to be equivalence relation if it is reflexive, symmetric and transitive in nature

Hence the relation is an equivalence relation

b) A relation is said to be reflexive if (a,a) belongs to R

s1 R s1 -> Since s1 can be rotated or flipped,horizontal,diagonal in order to obtain again s1

hence the relation is reflexive in nature

A relation is said to be symmetric if (a,b) belongs to R, then (b,a) belongs to R

if s1 R s2=> implies s1 can be rotated or flipped,horizontal,diagonal to form s2

Then s2 R s1= > it will also be true, since we can also move s2 to form s1

A relation is said to be transitive if (a,b) and (b,c) belongs to R, then (a,c) belongs to R

s1 R s2 => s1 can be rotated or flipped,horizontal,diagonal to form s2

s2 R s3 => s2 can be rotated or flipped,horizontal,diagonal to form s3

s1 R s3 => (rotated or flipped,horizontal,diagonal s1 to obtain s2) + (Now rotated or flipped,horizontal,diagonal modified s2 to obtain s3) which is possible from the second

Hence s1 R s3 is possible

Therefore, relation is reflexive,symmetric and transitive

A relation is said to be equivalence relation if it is reflexive, symmetric and transitive in nature

Hence the relation is an equivalence relation

Equivalence classes will be Rotation classes, Flipped against horizontal, Flipped against vertical or Flipped against diagonal