List all elements of the relations

Question

The task is to List all elements of the relations (d) S`, S-1 and S ₀T.

S ⊆{1, 2, 3, 4, 5} x {1, 2, 3, 4, 5}
S = {〈1, 1〉, 〈1, 3〉, 〈1, 5〉, 〈2, 2〉, 〈2, 4〉, 〈3, 1〉, 〈3, 3〉, 〈5, 3〉}

My solution is:

S-1 = {〈1,1〉, 〈3,1〉, 〈5,1〉, 〈2,2〉, 〈4,2〉, 〈1,3〉, 〈3,3〉, 〈3,5〉}

S ₀T = {〈1,3〉, 〈1,5〉, 〈1,1〉, 〈2,4〉, 〈3,1〉, 〈3,3〉, 〈3,5〉, 〈5,1〉}

But I did not get the idea how to do S`. Can please somebody help?

Answer 1

Let $U = \{1, 2, 3, 4, 5\} \times \{1, 2, 3, 4, 5\}$.

$S-1 = \{\langle b, a \rangle \in U: \langle a, b \rangle \in S\}$,

$S \circ T = \{\langle a, c\rangle \in S ~~|~~ \exists b: \langle a,b \rangle \in T\land \langle b,c \rangle \in S\}$,

$S' = \{\langle a,b \rangle \in U : \langle a,b \rangle \not\in S\}$.

The task is to calculate 1. ($S-1$), 2. ($S \circ \top$), and 3. ($S'$) for the following set:

$S = \{〈1, 1〉, 〈1, 3〉, 〈1, 5〉, 〈2, 2〉, 〈2, 4〉, 〈3, 1〉, 〈3, 3〉, 〈5, 3〉\}$

1. $(S-1) = \{\langle b, a \rangle \in S: \langle a, b \rangle \in S\}$ $= \{\langle 1,1\rangle, \langle 3,1\rangle, \langle 5,1\rangle, \langle 2,2\rangle, \langle 4,2\rangle, \langle 1,3\rangle, \langle 3,3\rangle, \langle 3,5\rangle\} ~{\color{green}\checkmark}$

2. $(S\circ \top) = \{\langle a, c\rangle \in S ~~|~~ \exists b: \langle a,b \rangle \in T\land \langle b,c \rangle \in S\}$
   $= \{\langle 1,1\rangle, \langle 1,3\rangle, \langle 1,5\rangle, \langle 2,2\rangle, \langle 3,1\rangle, \langle 3,3\rangle, \langle 5,3\rangle\}$.

3. $S'$ = (see the first comment below).