Consider binary relationsR, S, T. Prove the following:

(a)(RS)¹=R¹S¹.

(b)(RS)¹=R¹S¹.

(c)T(RS)=(TR)(TS).

(d)T(RS)(TR)(TS).

Consider binary relationsR, S, Tand setsX, Y. Prove the following:

(a)(SR)[X] =S[R[X]].

(b)R[XY]=R[X]R[Y].

(c)XR¹[R[X]].

(d)AssumeRis a function. Prove thatR¹[XY] =R¹[X]R¹[Y].

(e)Consider a setAand the binary relationRAP(A) defined by

x,XRxX

GivenX,YP(A), calculateR¹[{X}] andR¹[{X, Y}].