📘 set theory

1. The problem asks for the number of proper subsets of the set $L = \{\text{Bill Gates}, \text{Warren Buffett}, \text{Larry Ellison}, \text{Jeff Bezos}, \text{Charles Koch}, \text

1. **Stating the problem:** We are given sets $B_k = \left[ \frac{5}{k}, \frac{5k+2}{k} \right]$ for $k=1$ to $5$ with universal set $\mathbb{R}$.

1. **State the problem:** We are given sets $B_k$ for $k=1$ to $5$ and need to find three things: the union $\bigcup_{k=1}^5 B_k$, the intersection $\bigcap_{k=1}^5 B_k$, and the s

1. **Problem Statement:** Happy Elementary has three clubs: Art (A), Book (B), and Computer (C) with given membership counts and intersections. We need to analyze membership counts

1. **Problem Statement:** We have a survey of 50 people about their lunch preferences among three hotels: Hilltop, Serena, and Lemigo. Given are the total numbers who ate at each h

1. **State the problem:** We have three sets defined on the universal set $\xi = \{x : x \text{ is a positive integer less than } 16\}$.

1. **State the problem:** We are given four pairs of sets (a), (b), (c), and (d). For each pair, we need to determine the relationship between the sets using the options: - Equival

1. **Problem Statement:** Determine if each given set is finite or infinite. If finite, find its cardinality. 2. **Set (a):** $\{1, 2, 3, \ldots, 30\}$

1. **State the problem:** We have a Venn diagram with three sets: S (sheep), C (cows), and P (pigs). The numbers inside represent farms raising those animals. We want to find how m

1. **State the problem:** We have three sets: - Universal set $\xi = \{x : x \text{ is an integer and } 1 \leq x \leq 6\}$

1. **State the problem:** We need to find the number of elements in the intersection of sets $K$, $L$, and $M$, where: - $K$ is the set of multiples of 3.

1. **Problem 1:** Given sets:

1. **Problem 3:** Given a Venn diagram with universal set $\varepsilon = 350$, $n(C) = 200$ (families owning a car), $n(M) = 120$ (families owning a motor-cycle), and subsets label

1. **State the problem:** We need to place the numbers 28, -43, 19, -8.09, and 4/5 into a Venn diagram with three nested sets: Whole numbers inside Rational numbers inside Integers

1. The problem is to understand the symbol \(\emptyset\), which represents the empty set in mathematics. 2. The empty set is a set that contains no elements.

1. The problem asks to list all subsets of the set $\{1, 2\}$.\n\n2. A subset is any combination of elements from the original set, including the empty set and the set itself.\n\n3

1. The problem asks whether the statement $\{0\} = \emptyset$ is true or false. 2. Recall that $\emptyset$ is the empty set, which means it contains no elements.

1. The problem asks to find the value of $n(A \cup B)$ given $n(A) = 7$, $n(B) = 12$, and $n(A \cap B) = 6$. 2. The formula for the union of two sets is:

1. **State the problem:** We have two sets:

1. The problem asks to interpret the expression $B' \cup T'$ in the context of pupils playing basketball (B), cricket (C), and tennis (T). 2. Here, $B'$ represents the set of pupil

1. **State the problem:** We are given universal set $U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\}$ and subsets $A = \{1, 2, 3, 4, 5, 10\}$, $B = \{3, 4, 6, 8, 10\}$, and $C = \{4