10. Translate the following into MPL (not into PL):Some dogs are kelpies, some are beagles, and the rest are corgis.

To translate the given statement into Mathematical Predicate Logic (MPL), we will break down the components of the sentence and represent them using predicates.

Step 1: Identify the Components

1. Terms:

   • Dogs
   • Kelpies
   • Beagles
   • Corgis

2. Predicates:

   • Let $D(x)$ represent "x is a dog."
   • Let $K(x)$ represent "x is a kelpie."
   • Let $B(x)$ represent "x is a beagle."
   • Let $C(x)$ represent "x is a corgi."

Step 2: Structure the Statement

The statement "Some dogs are kelpies, some are beagles, and the rest are corgis" can be interpreted in parts as follows:

• There exists some dogs that are kelpies.
• There exists some dogs that are beagles.
• The rest of the dogs must be corgis.

Step 3: Formulate the MPL

Using the existential quantifier, we will express the statement in MPL:

• There exist dogs that are kelpies: $\exists x (D(x) \land K(x))$
• There exist dogs that are beagles: $\exists x (D(x) \land B(x))$
• The rest are corgis:
  • For this part, we can express that if a dog is not a kelpie or a beagle, it is a corgi: $\forall x (D(x) \land \neg K(x) \land \neg B(x) \rightarrow C(x))$

Final MPL Representation

Combining these parts, the full representation in MPL is: $\exists x (D(x) \land K(x)) \land \exists y (D(y) \land B(y)) \land \forall z (D(z) \land \neg K(z) \land \neg B(z) \rightarrow C(z))$

This represents the original statement accurately in Mathematical Predicate Logic.