Determine the validity of the next arguments by using Euler circles, then translate the statements into logical statements using basic connectives, and using truth tables, determine the validity of the arguments. Compare your answers.
(a)No A is B
(b) All B is A
Some C is A All C is A______
Some Cis not B
All C is B
To determine the validity of the arguments using Euler circles, we can draw circles to represent the sets involved in the statements and check if the statements hold true within the circles.
(a) No A is B:
To represent this statement, we can draw two non-overlapping circles, one for A and one for B. If there is no overlap between the two circles, this indicates that no element of A is also an element of B. Therefore, the statement is valid.
(b) All B is A:
This statement indicates that all elements of set B are also elements of set A. We can represent this by drawing a circle for B completely contained within the circle for A. This indicates that all B is indeed A, so the statement is valid.
Some C is A:
To represent this statement, we can draw circles for both C and A with an overlapping region. This indicates the existence of at least one element that belongs to both C and A. Therefore, the statement is valid.
All C is A:
This statement suggests that all elements of C are included in the set A. We can represent this by drawing a circle for C completely contained within the circle for A. This indicates that all C is indeed A, so the statement is valid.
Some C is not B:
To represent this statement, we can draw circles for both C and B with a non-overlapping region. This indicates the existence of at least one element of C that does not belong to B. Therefore, the statement is valid.
All C is B:
This statement indicates that all elements of C are also elements of B. We can represent this by drawing a circle for C completely contained within the circle for B. This indicates that all C is indeed B, so the statement is valid.
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Now, let's translate these statements into logical statements using basic connectives:
(a) No A is B:
This can be translated as "For every x, if x is an element of A, then x is not an element of B."
(b) All B is A:
This can be translated as "For every x, if x is an element of B, then x is an element of A."
Some C is A:
This can be translated as "There exists at least one x that is an element of both C and A."
All C is A:
This can be translated as "For every x, if x is an element of C, then x is an element of A."
Some C is not B:
This can be translated as "There exists at least one x that is an element of C but not an element of B."
All C is B:
This can be translated as "For every x, if x is an element of C, then x is an element of B."
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To determine the validity of these logical statements, we can use truth tables. A truth table lists all possible combinations of truth values for the variables involved and evaluates the resulting truth value of the statement for each combination.
Comparing the Euler circles and the truth tables, we can see that the validity of the arguments matches in both cases. All statements that were valid in the Euler circles interpretation are also valid in the logical statements interpretation.