1. Explain whether the following statement is a valid definition: “A 150° angle is an obtuse angle.” Use the converse, biconditional, and at least one Euler diagram to support your answer.

Question

2. Explain the purposes of inductive and deductive reasoning in mathematics. Be sure to define both inductive reasoning and deductive reasoning and describe how each can help you develop and prove theorems.

Answer 1

“A 150° angle is an obtuse angle.”

We will represent the proposition as follows:
p=the angle is 150°
q=it is an obtuse angle

The above proposition (may or may not be true) is equivalent to:
p->q (If the angle is 150°, it is an obtuse angle).

The converse is
q->p (If an angle is obtuse, it is 150°)

The biconditional is:
p<->q (If the angle is 150°, it is an obtuse angle, and if an angle is obtuse, it is 150°)

We can see that p->q is true, but q->p is not. Consequently p<->q is not true, because
p<->q ≡ p->q ∧ q->p,
so if q->p is false, p<->q is also false.

The Euler diagram for p->q is a small circle P completely inside a bigger circle Q, so that whenever p is true, q has to be true.

Try to draw the Euler diagram for the other two cases.