4. Let T be a square with each side having length 1 located in the plane so that the topside is parallel to the x-axis. Let S be the set of coloured squares obtainable from T bypainting each side with one of the colours red and blue. Any combination of colours isallowed, 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 thebottom edge being red and all others blue.(a) Define a relation R on S by s1 R s2 if and only if s1 can be rotated in place sothat the rotated coloured square is identical to s2. Prove that R is an equivalencerelation and find the 6 equivalence classes. (Your sets can contain pictures of thecoloured squares they contain, or you can use abbreviations like RRBR to denotethe side colours going clockwise from the top.)(b) Define a relation F on S by s1 F s2 if and only if s1 can be rotated or flippedover a vertical, horizontal or diagonal line so that the resulting coloured squareis identical to s2. Taking for granted that this is an equivalence relation, find thepartition of S induced by the equivalence classes.

5. Let be an equivalence relation on the set A = {1, 2, . . . , 8}, and denote the equivalenceclass of x A by [x].(a) Suppose that 1 [3], 4 [2], and 2 [1]. Prove that [4] = [3].(b) Ignore part (a) and suppose now that has 3 equivalence classes. If [1] has 2elements, [2] has 3 elements, 1 6, 2 5, and 7 5, theni. Write as a set of ordered pairs.ii. Find the partition of A induced by

.6. Ten students attend a games night at which they willcompete in teams of two. How many different collections of 5 such teams are there?

Please help I've been trying to solve this for hours