Some of the following arguments are valid, whereas others exhibit the converse or inverse error. Introduce symbols (using “Let” statements) and use them to write the logical form of each argument. State whether the argument is valid or invalid. If the argument is valid, identify the rule of inference from §2.3 that guarantees its validity. If the argument is invalid, state whether the converse or the inverse error is made. (a) If Jane is an actuarial math major, then Jane is required to take MATH 21001 . Jane is not an actuarial math major. ∴ Jane is not required to take MATH 21001. (b) If this number is divisible by 12, then it is divisible by 6. If this number is divisible by 6, then it is divisible by 2. ∴ If this number is divisible by 12, then it is divisible by 2. (c) If this function is continuous on the interval [5, 13], then it is integrable on [5, 13]. This function is not integrable on [5, 13]. ∴ This function is not continuous on [5, 13]. (d) If this function is differentiable on the interval (0, 2), then it is continuous on (0, 2). This function is continuous on (0, 2). ∴ This function is differentiable on (0, 2) (e) Jonah is from Kent or Jonah is from Akron. If Jonah is from Kent, then Jonah is from Ohio. If Jonah is from Akron, then Jonah is from Ohio. ∴ Jonah is from Ohio.
Some of the following arguments are valid, whereas others exhibit the converse or inverse error. Introduce symbols (using “Let” statements) and use them to write the logical form of each argument. State whether the argument is valid or invalid. If the argument is valid, identify the rule of inference from §2.3 that guarantees its validity. If the argument is invalid, state whether the converse or the inverse error is made.
(a) If Jane is an actuarial math major, then Jane is required to take MATH 21001 .
Jane is not an actuarial math major.
∴ Jane is not required to take MATH 21001.
(b) If this number is divisible by 12, then it is divisible by 6.
If this number is divisible by 6, then it is divisible by 2.
∴ If this number is divisible by 12, then it is divisible by 2.
(c) If this function is continuous on the interval [5, 13], then it is
This function is not integrable on [5, 13].
∴ This function is not continuous on [5, 13].
(d) If this function is differentiable on the interval (0, 2), then it is continuous on (0, 2).
This function is continuous on (0, 2).
∴ This function is differentiable on (0, 2)
(e) Jonah is from Kent or Jonah is from Akron. If Jonah is from Kent, then Jonah is from Ohio.
If Jonah is from Akron, then Jonah is from Ohio.
∴ Jonah is from Ohio.
Solution:
(a) Let us use the following symbols to represent the following:
Am : Jane is Actuarial math major
M : Jane is required to take MATH 21001
Now, the given set of statements can be expressed using the symbols as,
(Am --> M) & ~Am --> ~M
Now this is actually not true, and it leads to inverse error.
(b) Let us use the following symbols:
d12 : A number is divisible by 12
d6 : A number is divisible by 6
d2 : A number is divisible by 2
Now the given set of statements can be expressed using the symbols as,
(d12 --> d6) & (d6 --> d2) --> (d12 --> d2)
This statement is valid by the rule of inference called "Hypothetical Syllogism"
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