Problem discussed in Relation in Set Theory

Question

I am reading a book Axioms and Set Theory - A first course in Set Theory by Robert Andr´e

In the book an example if discussed regarding composition of two relations R and T

For a set S

S = {a,b,c,{a},{a,b},{{c}},∅,{∅}}

The relations R and T on S are given as

R = {(x, y) : x ∈ y}

T = {(x, y) : x = {y}}

So now,

R = { (a, {a}), (a, {a, b}), (b, {a, b}), (∅, {∅}) }

T = { ({a}, a}), ({∅},∅})}

We define T∘R

T∘R = {(x, y) : (z, y) ∈ T for some z ∈ imR}

For the relation T∘R we obtain:

T∘R = {(a, a), (∅,∅)}

For the relation R∘T we obtain:

R∘T = {({a}, {a}), ({∅}, {∅})}

But I am getting my answer as:

R∘T = {({a}, {a}), ({a}, {a, b}), ({∅}, {∅})}

Am I making any mistake or the example discussed in book has the error

Answer 1

I think your definition of $T\circ R$ needs a bit of work, it should read

$$ T\circ R = \{(x,y)\mid \text{there exists $z$ such that }(x,z)\in R\text{ and }(z,y)\in T\} $$

Note: it matters which $x$ the $z$ comes from in the relation $R$.

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Other than that, I would agree with your answer: $(\{a\},\{a,b\})$ should be in $R\circ T$ since we have $(\{a\},a)\in T$ and $(a,\{a,b\})\in R$.