Math Problem Statement

Which of the following are true and which are false? (a) 10 ∈/ (−∞, π2 ). (b) π ∈ (2, ∞).

Solution

Let's break down each statement:

Statement (a) 10/(,π2)10 \in/ (-\infty, \frac{\pi}{2})

  • The notation 10/(,π2)10 \in/ (-\infty, \frac{\pi}{2}) means that 10 is not an element of the interval (,π2)(-\infty, \frac{\pi}{2}).
  • The interval (,π2)(-\infty, \frac{\pi}{2}) represents all real numbers less than π2\frac{\pi}{2} (about 1.57).
  • Since 10 is clearly greater than π2\frac{\pi}{2}, it does not belong to this interval.

So, Statement (a) is true. 10(,π2)10 \notin (-\infty, \frac{\pi}{2}).

Statement (b) π(2,)\pi \in (2, \infty)

  • The notation π(2,)\pi \in (2, \infty) means that π\pi is an element of the interval (2,)(2, \infty).
  • The interval (2,)(2, \infty) represents all real numbers greater than 2.
  • Since π3.14159\pi \approx 3.14159, which is indeed greater than 2, it belongs to this interval.

So, Statement (b) is true. π(2,)\pi \in (2, \infty).

Final Answer:

  • (a) True
  • (b) True

Would you like more details on these statements, or do you have any questions about intervals or set notation?

Related Questions:

  1. What is the meaning of \in and \notin in set theory?
  2. How do open and closed intervals differ, and what does that affect?
  3. Can you explain the difference between intervals like (,a)(-\infty, a) and (a,)(a, \infty)?
  4. How do you convert radians to degrees, and vice versa, when working with intervals?
  5. What is the significance of π\pi in mathematics and its approximations?

Tip: Always remember that (,a)(-\infty, a) means numbers less than aa, and (a,)(a, \infty) means numbers greater than aa. If you're unsure, just think of the number line!