这是一份nottingham诺丁汉大学MATH1057作业代写的成功案例
Let $A$ be the set of people, and $P, F, M, S, B$ the relations over $A$ of ‘parent of’, ‘father of’, ‘mother of’, ‘sister of’ and ‘brother of’ respectively. Describe exactly the following relative products. (a) $P \circ P$, (b) $M \circ F$, (c) $S \circ P$, (d) BoB. Warnings: (1) Be careful about order. (2) In some cases there will be a handy word in English for just the relation, but in others it will have to be described in a more roundabout (but still precise) way.
PoP $P$ ‘grandparent of’. Reason: $a$ is a grandparent of $c$ iff there is an $x$ such that $a$ is a parent of $x$ and $x$ is a parent of $c$.
$M \circ F=$ ‘maternal grandfather of’. Reason: $a$ is a maternal grandfather of $c$ iff there is an $x$ such that $a$ is a father of $x$ and $x$ is a mother of c. Comments: Two common errors here. (1) Getting the order wrong. (2) Rushing to the answer ‘grandfather of’, since a father of
MATH1057 COURSE NOTES ：
Let $f: A \rightarrow B$, i.e. let $f$ be a function from $A$ to $B$, and let $X \subseteq A$. The image under $f$ of $X \subseteq A$ is the set ${b \in B: \exists a \in X, b=f(a)}$, which can also be written more briefly as ${f(a): a \in A}$. Thus, to take limiting cases as examples, the image $f(A)$ of $A$ itself is range $(f)$, and the image $f(\varnothing)$ of $\varnothing$ is $\varnothing$, while the image of a singleton subset ${a} \subseteq A$ is the singleton ${f(a)}$. Thus image is not quite the same thing as value: the value of $a \in A$ under $f$ is $f(a)$, while the image of ${a} \subseteq A$ under $f$ is ${f(a)}$. However, many texts also use the term ‘image’ rather loosely as a synonym of ‘value’. Always check what your author means.