Final answer:
The original conditional statement, represented as 'if the dimensions of a rectangle are doubled, then the area increases by a factor of four' is symbolically p → q, making option a) correct. The inverse, which is '~p → ~q', states that if the dimensions are not doubled, the area doesn't increase by a factor of four, making option b) correct.
Step-by-step explanation:
The question revolves around understanding the logical propositions related to the geometric statement that doubling the dimensions of a rectangle increases the area by a factor of four. The original conditional statement is "if the dimensions of a rectangle are doubled (p), then the area increases by a factor of four (q)". This can be represented symbolically as p → q.
The inverse of the original statement would negate both the hypothesis and the conclusion, leading to "if the dimensions of a rectangle are not doubled (~p), then the area does not increase by a factor of four (~q)". Symbolically, this is represented as ~p → ~q, which is option b). The converse of the original statement switches the hypothesis and conclusion, leading to "if the area of a rectangle increase by a factor of four (q), then the dimensions of the rectangle were doubled (p)". This is represented as q → p, but this is not the original conditional statement; it's the converse.
The contrapositive of the original statement negates and switches the hypothesis and conclusion, resulting in "if the area of a rectangle does not increase by a factor of four (~q), then the dimensions were not doubled (~p)". Symbolically, this is represented as ~q → ~p, which is not mentioned as an option in the question. Hence, the correct options that represent the original conditional statement and its inverse are options a) and b) respectively.