Do my relational algebra operations produce the desired results?
This is indeed the error:


    R4 := R2 &#x2229; R3 


If we try to get the intermediate results of your solution, we get: 

R1:


bid  bname      color       sid  date
---  ---------  -----       ---  ----------
102  interlake  red         31   8/11/2014
103  clipper    green       22   7/05/2014
103  clipper    green       58   8/11/2014
104  clipper    red         22   8/10/2014


and then:

R2: (only the 'red')


bid  bname      color       sid  date
---  ---------  -----       ---  ----------
102  interlake  red         31   8/11/2014
104  clipper    red         22   8/10/2014


R3: (only the 'green')


bid  bname      color       sid  date
---  ---------  -----       ---  ----------
103  clipper    green       22   7/05/2014
103  clipper    green       58   8/11/2014


R4 will be an empty relation because nothing can be both red and green. Of course that is not what the exercise asks for.

R4:


bid  bname      color       sid  date
---  ---------  -----       ---  ----------




What you need to do is to get R2' and R3', the projections of R2 and R3 respectively, with only the attribute sid. Then take their intersection for R4:


    R1  := ...
    R2  := ...
    R3  := ...
    R2' := πsid(R2)
    R3' := πsid(R3)
    R4  := R2' &#x2229; R3'
    R5  := ...
    R6  := ...