📘 set theory

1. **Problem:** Determine if the relation $R = \{(A,B) \in P \times P : A = B\}$ on the power set $P$ of $S = \{2,3,7\}$ is reflexive, symmetric, and/or transitive. 2. **Recall def

1. **State the problem:** We have 200 tennis enthusiasts surveyed about attendance at three grand slam tournaments: Australian Open (A), Wimbledon (B), and US Open (C). Given atten

1. **Rewrite the set** $A = \{x \mid x \in \mathbb{Z}, -5 < x < -2\}$ **in roster form.** The set $A$ contains all integers $x$ such that $x$ is greater than $-5$ and less than $-2

1. **State the problem:** We have 12 pupils in a class. They are asked if they have a brother or a sister. 2. **Given data:**

1. **State the problem:** We have 12 pupils in a class (set $\varepsilon$). Among them, 9 have a brother (set $B$), 7 have a sister (set $S$), and 2 have neither. 2. **Goal:** Find

1. **State the problem:** We have universal set $U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$, sets $A = \{3, 5, 7, 10\}$ and $B = \{2, 3, 5, 8, 10\}$. We want to find $(A \cap B)'$ usin

1. The problem is to find the union of three sets: $A$, $B$, and $C$, specifically $(A \cup B) \cup C$. 2. The union of two sets $X$ and $Y$, denoted $X \cup Y$, is the set contain

1. The problem asks to find all numbers in the set $Q \cap R'$ where $Q = \{4, 8, 12, 16, 20, 24\}$ and $R = \{6, 12, 18, 24, 30\}$. Here, $R'$ denotes the complement of $R$ relati

1. **State the problem:** We are given two sets: - $Q = \{\text{positive multiples of } 3 \text{ that are less than } 10\}$

1. The problem is to understand what a subset is in mathematics. 2. A subset is a set where every element in it is also an element of another set.

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.