Considering the following statements, what is $X \rightarrow Z$?
$X \rightarrow Y$ : If the sum of the interior angles of a shape is $180^{\circ}$, then it's a triangle.
$Y \rightarrow Z$ : If a shape is a triangle, then it has three sides.
A) $X \rightarrow Z$ : If a shape has three sides, then the sum of the interior angles of the shape is $180^{\circ}$.
B) $X \rightarrow Z$ : If the sum of the interior angles of a shape isn't $180^{\circ}$, then it doesn't have three sides.
C) $X \rightarrow Z$ : If the sum of the interior angles of a shape is $180^{\circ}$, then it has three sides.
D) $X \rightarrow Z$ : If a shape has three sides, then it's a triangle.
Answer (1)
Analyze the given conditional statements X → Y and Y → Z .
Apply the transitive property of logical implication to deduce X → Z .
Determine that X → Z : If the sum of the interior angles of a shape is 18 0 ∘ , then it has three sides.
Select the matching option from the given choices: C .
Explanation
Analyze the given statements. We are given two conditional statements:
X → Y : If the sum of the interior angles of a shape is 18 0 ∘ , then it's a triangle.
Y → Z : If a shape is a triangle, then it has three sides.
We want to find the logical implication X → Z .
Apply the transitive property. The transitive property of logical implication states that if X → Y and Y → Z , then X → Z .
Applying this property to the given statements, we have:
X → Z : If the sum of the interior angles of a shape is 18 0 ∘ , then it has three sides.
Compare with the given options. Now, we compare our derived implication with the given options:
A) X → Z : If a shape has three sides, then the sum of the interior angles of the shape is 18 0 ∘ .
B) X → Z : If the sum of the interior angles of a shape isn't 18 0 ∘ , then it doesn't have three sides. C) X → Z : If the sum of the interior angles of a shape is 18 0 ∘ , then it has three sides. D) X → Z : If a shape has three sides, then it's a triangle.
Option C matches our derived implication.
State the final answer. Therefore, the correct answer is:
X → Z : If the sum of the interior angles of a shape is 18 0 ∘ , then it has three sides.
Examples
In architecture, understanding logical implications helps ensure structural integrity. For instance, if a design requires a triangular support ( X → Y ), and triangular shapes inherently have three sides ( Y → Z ), then the design implicitly relies on a three-sided structure whenever a 18 0 ∘ angle sum is specified ( X → Z ). This ensures that the intended structural properties are maintained throughout the design process.
