📘 set theory

1. **Problem Statement:** Solve question 11 (not provided in the prompt). Since question 11 is missing, I cannot solve it directly. 2. **Explanation:** To solve any set theory prob

1. **Problem Statement:** Find the specified sets and their operations for questions 9 and 10. ---

1. **State the problem:** Find the intersection and union of sets \(M = \{1, 3, 5, 7, 9, 11, 13, 15, 17, 19\}\) and \(N = \{3, 6, 9, 11, 13\}\). 2. **Recall definitions:**

1. The problem states the distributive law of set operations: $$p \cap (q \cup r) = (p \cap q) \cup (p \cap r)$$. 2. This law means the intersection of set $p$ with the union of se

1. **State the problem:** We need to find the complement of set $M$ with respect to the universal set $U$, denoted as $M'$. The complement $M'$ consists of all elements in $U$ that

1. The problem asks to find the intersection of sets $M$ and $Z$, denoted as $M \cap Z$. 2. The intersection of two sets contains all elements that are common to both sets.

1. **State the problem:** We are given the total number of elements in the universal set $\xi$ as $n(\xi) = 21$, the number of elements in the union of sets $A$ and $B$ as $n(A \cu

1. The problem states that we have three sets $A$, $B$, and $K$ such that $A \subset K$, $B \subset K$, and $A \cap B = \emptyset$. 2. This means both $A$ and $B$ are subsets of $K

1. The problem is to understand the set notation $A = \{x \in \mathbb{N} : -2 \leq x\}$ and determine what elements belong to this set. 2. Here, $\mathbb{N}$ represents the set of

1. **Problem statement:** In a class of 50 students, 30 like mango, 25 like guava, and 10 like none of the fruits. Find the number of students who like both fruits, mango only, and

1. **Problem Statement:** Describe the shaded regions in the Venn diagram using set notation.

1. **Problem Statement:** We have a survey about people liking apricots (A), bananas (B), and cantaloupes (C) with given counts and intersections. We want to find the number who li

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

1. **State the problem:** We have 120 students studying Mathematics (M), Economics (E), and Computer Studies (C) with given overlaps. We want to represent this information using a

1. **State the problem:** We have 120 students surveyed about their study of Mathematics (M), Economics (E), and Computer Studies (C). Given the numbers studying each subject and t

1. The problem asks for two difficult problems involving sets. 2. Let's consider the first problem: Given two sets $A$ and $B$, find the set expression for elements that are in $A$

1. **Problem statement:** In a class of 50 students, 30 like math, 25 like science, and 10 like neither. We need to find:

1. **Problem statement:** In a class of 30 students, 30 like Math, 25 like Science, and 10 like neither subject. We need to find: - Number of students who like both subjects.

1. **State the problem:** Determine whether each set C is a subset of set D for the given pairs. 2. **Recall the definition of subset:** A set $C$ is a subset of $D$, written $C \s

1. **Stating the problem:** We have data about 120 students studying three languages: French (F), German (G), and Russian (R). We want to organize this data and understand the numb

1. **Problem statement:** Prove that the complement of the union of sets $A$ and $B$ equals the intersection of their complements, i.e., $$\overline{A \cup B} = \overline{A} \cap \