= Solution
The <core of a cooperative game> consists of efficient allocations satisfying $\sum_{i\in S}x_i\geq v(S)$ for every <coalition>, so no <coalition> can obtain a greater total payoff by leaving. Efficiency has already been checked. For singletons,
$$
\frac83\geq0,\qquad\frac{14}3\geq2,\qquad\frac{23}3\geq5.
$$
For pairs,
$$
\frac{22}3\geq6,\qquad\frac{31}3\geq9,\qquad\frac{37}3\geq11.
$$
The empty constraint is automatic and the grand constraint is equality. Thus every core inequality holds and
$$
\boxed{x\in\operatorname{core}(v):\quad\text{(d) is true.}}
$$
All four assertions about this allocation are true, although their definitions impose different requirements.
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