Sure, let's break down these conditional statements and produce the required logical forms. We'll also determine the truth values for each of the statements.
Statement 7: Two planes intersect at a line.
1. Conditional: If-then form
- If two planes intersect, then they intersect at a line.
- True: This is a true statement because the intersection of two planes in three-dimensional geometry is indeed a line, provided that the planes are not parallel or coincident.
2. Converse:
- If two entities intersect at a line, then they are two planes.
- False: This is a false statement. While two planes intersect at a line, other geometrical entities can also intersect in such a configuration, and not all intersections at a line involve two planes.
3. Inverse:
- If two planes do not intersect, then they do not intersect at a line.
- True: This is true because if two planes do not intersect at all, they cannot intersect at a line.
4. Contrapositive:
- If two entities do not intersect at a line, then they are not two planes.
- False: This statement is misleading because it suggests that the only way for entities to be planes is to intersect at a line, which is not the case. Two planes can also be parallel and not intersect at all.
5. Biconditional:
- Two planes intersect if and only if they intersect at a line.
- True: This statement accurately describes the relationship between two intersecting planes in three-dimensional space.
Statement 8: A relation that pairs each input with exactly one output is a function.
1. Conditional: If-then form
- If a relation pairs each input with exactly one output, then it is a function.
- True: This is the definition of a function. Each input (or domain element) is associated with exactly one output (or range element).
2. Converse:
- If a relation is a function, then it pairs each input with exactly one output.
- True: This is the definition of what it means to be a function. Therefore, the converse is true for this statement.
3. Inverse:
- If a relation does not pair each input with exactly one output, then it is not a function.
- True: This is consistent with the definition of a function. If a relation does not have the property of pairing each input with exactly one output, it is not a function.
4. Contrapositive:
- If a relation is not a function, then it does not pair each input with exactly one output.
- True: The contrapositive of a true statement is always true. This restates the definition in a logically equivalent manner.
5. Biconditional:
- A relation pairs each input with exactly one output if and only if it is a function.
- True: This biconditional statement accurately captures the definition of a function.
In summary, for statement 7, the conditional and its inverse are true, the converse and contrapositive are false and the biconditional is true. For statement 8, all the derived forms are true as they accurately describe the definition of a function.
