📘 set theory

1. **Problem:** Find the total number of elements in the power set of a set $A$ containing 15 elements. 2. **Formula:** The power set of a set with $n$ elements has $2^n$ elements.

1. **Problem:** Find the total number of elements in the power set of a set $A$ containing 15 elements. 2. **Formula:** The power set of a set with $n$ elements has $2^n$ elements.

1. **Stating the problem:** We want to prove De Morgan's law for sets, which states that the complement of the union of two sets $A$ and $B$ is equal to the intersection of their c

1. مسئله: تعداد اعضای مجموعه $[A-(B\cup C)]\cup B$ را بیابید که در آن $$A=\{1,4,5,7\},\quad B=\{3,4,6,8\},\quad C=\{1,3,5,7,9\}$$

1. مسئله: ثابت کنید $$A - (A \cap B) = A - B$$. 2. تعریف‌ها و فرمول‌ها:

1. பிரச்சினையை விளக்குதல்: இந்த வென்விருப்படத்தில் இரண்டு வட்டங்கள் உள்ளன: இடது வட்டம் "குடை உள்ள பிள்ளைகள்" மற்றும் வலது வட்டம் "பெண் பிள்ளைகள்".

1. The problem asks us to determine the correct relationship between two sets $A$ and $B$ among the options given: 2. Recall the definitions:

1. **Stating the problem:** We have two sets $A$ and $B$.

1. Тодорхойлолт: Олонлог нь бүх рациональ тоонууд $p$ ба $q$-ийн хослол бөгөөд $-2 \leq p \leq 4$, $p \neq 0$, мөн $q=2$ байна. 2. Олонлогийн нөхцөлүүдийг анхаарна: $p$ нь $-2$-оос

1. **Problem Statement:** Solve the first question: Distinguish between set intersection and set union.

1. **Problem statement:** Given sets with operations and elements:

1. مسئله را بیان می‌کنیم: باید مجموعه $A-B$ را پیدا کنیم که $A=\{x\in \mathbb{R} \mid -1-4 \leq x \leq -1-3\}$ و $B=\{n \in \mathbb{Q} \cap (-,f) \cap \sum n\}$ است. 2. ابتدا باید

1. **Problem Statement:** Identify which set operation corresponds to the shaded region in the given Venn diagram involving sets $A$, $B$, and universal set $U$. 2. **Recall Set Op

1. **Problem:** Find the union of sets $A$ and $B$. **Step 1:** Recall the union formula: $$A \cup B = \{x : x \in A \text{ or } x \in B\}$$

1. **Problem statement:** Given sets and their cardinalities, find possible elements for sets B, C, a value for x in the complement of A, and calculate $n(D \cup C)$. 2. **Recall d

1. **Problem (a):** Find the cardinality of the largest possible set $C$ such that $C \subseteq A$ and $C \subseteq B$ where $A=\{1,6,9,10,11\}$ and $B=\{2,5,7,8,12\}$. 2. **Step 1

1. **State the problem:** We need to shade the complement of the intersection of sets $C$ and $A$, denoted as $(C \cap A)'$. 2. **Recall definitions:**

1. **Problem Statement:** We have a survey of 200 students about reading three newspapers: Imvaho (I), New Times (N), and Daily Times (D). Given data:

1. **Problem statement:** We are given the number of students enrolled in commerce only, mathematics only, both commerce and mathematics, and others. We need to find the total numb

1. **Problem Statement:** We have a survey of 500 students with percentages liking football (F), hockey (H), and basketball (B). Given data:

1. **State the problem:** We need to verify and explain the set identities: