Truth tables are related to Euler circles. Arguments in the form of Euler circles can be translated into statements using the basic connectives and the negation as follows:

Let p be “The object belongs to set A.” Let q be “the object belongs to set B.”

All A is B is equivalent to p ------> q.

No A is B is equivalent to p ----> ~ q.

Some A is B is equivalent to p ^ q.

All A is not B is equivalent to p ^ ~ q.

Determine the validity of the next arguments by using Euler circles, then translate the states into logical statements using the basic connectives, and truth tables, determine the validity of the arguments. Compare and explain your answers.

(a) No A is B.

Some C is A.

Some C is not B.

(b) All B is A.

All C is A.

All C is B.

Let p be “The object belongs to set A.” Let q be “the object belongs to set B.”

All A is B is equivalent to p ------> q.

No A is B is equivalent to p ----> ~ q.

Some A is B is equivalent to p ^ q.

All A is not B is equivalent to p ^ ~ q.

Determine the validity of the next arguments by using Euler circles, then translate the states into logical statements using the basic connectives, and truth tables, determine the validity of the arguments. Compare and explain your answers.

(a) No A is B.

Some C is A.

Some C is not B.

(b) All B is A.

All C is A.

All C is B.

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