What are the two conditionals that form the statement below?
Two numbers are reciprocals if and only if their product is 1.
The two conditionals that form the statement are:
1. "If two numbers are reciprocals, then their product is 1."
2. "If the product of two numbers is 1, then they are reciprocals."
Is the definition of a straight angle seen below reversible? If yes, write it as a true biconditional.
A straight angle is an angle that measures 180°.
Yes, the definition of a straight angle is reversible. We can write it as a true biconditional:
An angle measures 180° if and only if it is a straight angle.
Is the following statement a good definition? Explain
A square is a figure with four right angles.
Yes, the statement "A square is a figure with four right angles" is a good definition.
A good definition should accurately describe the characteristics or properties of the object being defined. In this case, the statement clearly identifies the defining characteristic of a square, which is that it has four right angles. This definition effectively distinguishes squares from other shapes, such as rectangles, which may have four angles but not necessarily all right angles.
How can you write the statement "Obtuse angles have greater measure than acute angles" so that it is a good definition?
To write the statement "Obtuse angles have greater measure than acute angles" as a good definition, we need to include both the defining characteristic and the necessary condition.
A good definition should accurately describe the defining characteristic of the object in question and also capture the necessary condition that separates it from other objects.
A revised definition for this statement could be:
"An obtuse angle is an angle that measures more than 90 degrees, while an acute angle is an angle that measures less than 90 degrees."
This definition clearly states the defining characteristic of obtuse and acute angles, which is their respective measure in relation to 90 degrees. By providing this necessary condition, the definition highlights the distinction between obtuse and acute angles based on their measures.
A ligament is a band of tough tissue connecting bones or holding organs in place.
Write the second ligament definition above as a biconditional.
"A ligament is a band of tough tissue connecting bones or holding organs in place if and only if it is connecting bones or holding organs in place."
The statement "Two numbers are reciprocals if and only if their product is 1" can be broken down into two conditionals:
1. The first conditional is "If two numbers are reciprocals, then their product is 1." This is the implication that if two numbers are reciprocals, it guarantees that their product will be 1.
2. The second conditional is "If the product of two numbers is 1, then they are reciprocals." This is the converse of the first conditional, stating that if the product of two numbers is 1, it guarantees that they are reciprocals.
So, the two conditionals that form the given statement are:
1. If A and B are reciprocals, then A * B = 1.
2. If A * B = 1, then A and B are reciprocals.
To prove the statement "Two numbers are reciprocals if and only if their product is 1," you would need to prove both of these conditionals.