determine the validity of the next arguments by using Euler circles, then translate the statements into logical statements using the basic connectives, and using truth tables, determine the validity of the arguments. Compare 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
To determine the validity of the arguments using Euler circles, we can follow these steps:
1. Draw circles to represent the sets of A, B, and C.
2. For each statement, determine if it can be represented by overlapping regions in the circles.
Let's examine each argument:
(a) "No A is B"
Draw a circle to represent A and another circle to represent B. Since the statement says "No A is B," we would not have any overlapping region between the A and B circles. This representation would show the argument to be valid.
"Some C is A"
Draw a circle to represent C and another circle to represent A. Since the statement says "Some C is A," we would have an overlapping region between the C and A circles. This representation would show the argument to be valid.
"Some C is not B"
Draw a circle to represent C and another circle to represent B. Since the statement says "Some C is not B," we would have a region in the C circle that does not overlap with the B circle. This representation would show the argument to be valid.
Therefore, all three statements can be represented in Euler circles, and the argument is valid.
Now, let's translate the statements into logical statements using the basic connectives:
(a) "No A is B" can be translated as "For all x, if x is A, then x is not B."
"Some C is A" can be translated as "There exists an x, such that x is C and x is A."
"Some C is not B" can be translated as "There exists an x, such that x is C and x is not B."
Now we can use truth tables to determine the validity of the arguments:
To validate the argument, we compare the truth values of the premises (the translated statements) with the truth value of the conclusion. If the conclusion is true whenever all the premises are true, the argument is valid.
Now, let's create the truth tables:
For argument (a):
| A | B | C | No A is B | Some C is A | Some C is not B |
|---|---|---|-----------|-------------|-----------------|
| T | T | T | F | T | T |
| T | T | F | F | F | T |
| T | F | T | T | T | T |
| T | F | F | T | F | T |
| F | T | T | T | T | T |
| F | T | F | T | F | T |
| F | F | T | T | T | T |
| F | F | F | T | F | T |
All the premises (translated statements) are true for all combinations of truth values. Therefore, the argument is valid.
For argument (B):
| A | B | C | All B is A | All C is A | All C is B |
|---|---|---|------------|------------|------------|
| T | T | T | T | T | T |
| T | T | F | T | F | T |
| T | F | T | F | T | F |
| T | F | F | F | F | F |
| F | T | T | T | T | T |
| F | T | F | T | F | T |
| F | F | T | T | T | T |
| F | F | F | T | F | T |
All the premises (translated statements) are true for all combinations of truth values. Therefore, the argument is valid.
Comparing the results obtained from the Euler circles and the truth tables, we can conclude that both methods yield the same result, indicating that the arguments are valid.