Write the following in set notation:
(a) The set of all real numbers greater than 7 but less than 173.
(b) πd = 200 β 5π
(c) πs = β100 + 10π
(d) π· β π using the previous demand and supply equations
(a) The set can be represented in set notation as:
{π₯ | 7 < π₯ < 173}
Explanation: In this case, we use the interval notation to represent the set of real numbers between 7 and 173, excluding the endpoints. The symbol "|" represents "such that", and π₯ is the variable representing the real numbers within the specified range.
(b) The equation πd = 200 β 5π can be rewritten in set notation as:
πd = {200 β 5π₯ | π₯ β π
}
Explanation: The set πd represents the quantity demanded (πd) and is a function of the price (π). In set notation, we use the variable π₯ to represent the prices (which can be any real number), and π₯ β π
means π₯ belongs to the set of all real numbers.
(c) The equation πs = β100 + 10π can be written in set notation as:
πs = {β100 + 10π₯ | π₯ β π
}
Explanation: The set πs represents the quantity supplied (πs) and is also a function of the price (π). Similar to the previous equation, π₯ represents the price variable, and π₯ β π
means π₯ belongs to the set of all real numbers.
(d) The intersection of sets π· and π can be expressed in set notation as:
π· β π = {π₯ | π₯ β π
, π₯ β πd, π₯ β πs}
Explanation: When finding the intersection of sets π· and π, we look for values that satisfy all the given conditions. Here, π₯ represents the common elements between the sets π· and π. The notation π₯ β π
implies that π₯ is a real number, while π₯ β πd and π₯ β πs indicate that π₯ belongs to both sets πd and πs.