have you made any effort at it yet?
Let D and S be relations on A = {0, 1, 2, 3}.
• D = {(a, b) | b = (3a+1) mod 4}
• S = {(a, b) | b < a+1}
• D = {(0, 4), (1, 16), (2, 28), (3, 40)}
• S = {(1, 0), (2, 0), (2, 1), (3, 0), (3, 1), (3, 2), (4, 0), (4, 1), (4, 2), (4, 3)}
S_{°}D =???????????
Then find values of S_{°}D by writing all the intermediate steps
Either your first two definitions for D and S are wrong or the second two are. My guess is the second two are. Note: If , then , so it is not a relation on
Based on the first definition,
Similarly, the you gave does not match the definition you gave, nor is it even a relation on . Using the first definition,
Next, use the definition for composition of binary relations.
First the set $A$ has only four elements. Therefore, any relation can have at most sixteen pairs.
To do this question you must know what relations look like:
$D=\{(0,1)~,(1,0)~,(2,3)~,(3,2)\}$ and
$S=\{(0,0)~,(1,0)~,(1,1)~,(2,0)~,(2,1)~,(2,2)~,(3, 0)~,(3,1)~,(3,2)~,(3,3)\}$
Now, try again!
To calculate , take all of the pairs of and pairs of where the second coordinate from the pair in is the same as the first coordinate of the pair in and group them together:
Pairs in where the second coordinate is zero:
Pairs in where the first coordinate is zero:
Each pair from and that single pair from form pairs in :
(That is the first coordinate from the pair in and the second coordinate from the pair in )
Now, do the same for pairs in where the second coordinate is one and pairs in where the first coordinate is one. Then move on to when they are both two, and finally when they are both three. That will give you all elements of .