Determine whether these are valid arguments.
a. If x is a positive real number, then x2 is a positive real number. Therefore, if a2 is
positive, where a is a real number, then a is a positive real number.
b. Ifx2 ≠0,wherexisarealnumber,thenx≠0.Letabearealnumberwitha2 ≠0;then
a ≠ 0.
It would help if you proofread your questions before you posted them. Spaces?
Online "^" is used to indicate an exponent, e.g., x^2 = x squared.
a. This is a valid argument. It follows the logical rule of transitivity. If x is a positive real number and x^2 is a positive real number, then if a^2 is positive, a must also be a positive real number.
b. This is also a valid argument. It follows the contrapositive rule. If x^2 is not equal to 0, then x cannot be equal to 0. Similarly, if a^2 is not equal to 0, then a cannot be equal to 0.
To determine whether the given arguments are valid or not, we can use the rules of logic and reasoning to analyze their structure.
a. If x is a positive real number, then x^2 is a positive real number.
Therefore, if a^2 is positive, where a is a real number, then a is a positive real number.
To evaluate the validity of this argument, we can break it down into logical steps:
1. If x is a positive real number, then x^2 is a positive real number. (given)
2. Assume a^2 is positive. (assumption)
3. From step 2, we can infer that a^2 is greater than 0.
4. Since a^2 is greater than 0, it implies that a is not equal to 0, as the square of any nonzero number is positive.
5. Therefore, we can conclude that if a^2 is positive, then a is nonzero.
By analyzing the argument, we can see that it follows a valid logical structure. So, argument (a) is valid.
b. If x^2 ≠ 0, where x is a real number, then x ≠ 0.
Let a be a real number with a^2 ≠ 0; then a ≠ 0.
Let's evaluate this argument:
1. If x^2 ≠ 0, where x is a real number, then x ≠ 0. (given)
2. Assume a^2 ≠ 0. (assumption)
3. From step 2, we can infer that a^2 is not equal to 0.
4. Since a^2 is not equal to 0, it implies that a is also not equal to 0.
By examining the argument, we can see that it follows a valid logical structure. Therefore, argument (b) is also valid.
Both arguments have a logically valid structure and sound reasoning, so they are considered valid arguments.