Question

which of the following is a counterexample for this conditional statement? if a three-dimensional solid has six faces, then it is a cube. a cylinder a sphere a rectangular pyramid a rectangular prism

Explanation:

A counterexample for a conditional statement "If \( p \), then \( q \)" is an example where \( p \) is true (the hypothesis holds) but \( q \) is false (the conclusion does not hold).

1. Analyze each option:

• A cylinder: A cylinder has 2 circular faces and 1 curved surface, so it does not have six faces. Thus, the hypothesis (\( p \)) is false, so it cannot be a counterexample.
• A sphere: A sphere has no flat faces (it has a single curved surface), so the hypothesis (\( p \)) is false, so it cannot be a counterexample.
• A rectangular pyramid: A rectangular pyramid has a rectangular base and 4 triangular faces, so it has \( 1 + 4 = 5 \) faces. Thus, the hypothesis (\( p \)) is false, so it cannot be a counterexample.
• A rectangular prism: A rectangular prism (also known as a rectangular cuboid) has 6 faces (all rectangles, in general). However, a rectangular prism is not necessarily a cube (a cube is a special case of a rectangular prism where all faces are squares). So, the hypothesis (\( p \): "has six faces") is true, but the conclusion (\( q \): "is a cube") is false. This satisfies the definition of a counterexample.

Answer:

D. a rectangular prism