📘 set theory

1. The problem is to evaluate the empty set notation \( \left\{\;\right\} \), which represents the set with no elements. 2. By definition, an empty set contains no members; it is u

1. The problem is about understanding the sets \(A\) and \(B\).\n\n2. Set \(A = \{x: x \text{ is an odd natural number}\}\) includes all positive integers that are odd, such as 1,

1. **State the problem:** Given the set $$A = \{ \{-2, 2\}, \{-1, 1\}, 0 \}\), determine which of the following subset statements are true. 2. **Analyze Choice A:** $$\{ -2, 2 \} \

**Problem statement:** We are given a function $f: E \to F$, two subsets $A, B$ and subsets $A_1, A_2 \subseteq E$, $B_1, B_2 \subseteq F$. We need to prove:

1. **Show that if $A_1 \subset A_2$, then $f(A_1) \subset f(A_2)$:** By definition, $f(A) = \{y \in F \mid \exists x \in A, f(x) = y\}$. If $A_1 \subset A_2$, then every element $x

1. **State the problem:** Show using a Venn diagram and set theory that if $A \subset B$ then $B' \subset A'$, and conversely, if $B' \subset A'$ then $A \subset B$.

1. **State the problem:** We want to show through a Venn diagram and logical reasoning that if $A \subset B$ then $B' \subset A'$, and conversely, if $B' \subset A'$, then $A \subs

1. The problem states that 30 people were surveyed about playing badminton and cricket, with a Venn diagram representing the counts. 2. The Venn diagram numbers are:

1. The problem asks for the number of pizzas that have chillies on them based on the Venn diagram. 2. According to the Venn diagram, the pizzas that have chillies are the ones insi

1. First, solve the arithmetic expression given: 3 \times 16 \div 3 = ? and 24 \times 8 + ? \div 2. Step 1: Calculate 3 \times 16 = 48.

1. The problem asks to find the power set of the set $\{a,b\}$. 2. Recall, the power set of a set is the set of all its subsets, including the empty set and the set itself.

1. The problem asks us to find the power set of the set $\{a\}$. 2. The power set of a set is the set of all possible subsets of the original set, including the empty set and the s

1. The problem involves finding the cardinality $n(X \times Y)$ and related values from given cardinalities. 2. Recall the rule for Cartesian products cardinalities: $$n(X \times Y

1. The problem asks for the meaning of the set \(\{(x,y) : x \in A, y \in B\}\). This is by definition the Cartesian product \(A \times B\), the set of all ordered pairs where the

1. Problem statement: Given sets \(A=\{5, 10, 15, 20, 25\}\), \(B=\{5, 25, 125, 625\}\), and universal set \(U = A \cup B\), find: i) Elements of \(U\)

1. **State the problem:** We have two sets: - $A = \{x \mid x \text{ is an even number less than } 10\}$

1. Stating the problem: Given the set $$A = \{ \{ c, a, r \}, \text{taxi} \},$$ determine which of the following are true: (Choice A) $$a \in A$$

1. **State the problem:** We have a set $$A = \{1, \{2, 3\}, 4, 5\}$$ and need to determine which of these statements are true: - (A) $$\{2, 3\} \in A$$

1. **State the problem:** Given the set $$A = \{ \text{b, c, d}, \{ \text{a, e} \} \}$$, we need to determine which of the following statements are true.

1. Problem: Given the set $A = \{ 5, \{6, 7\}, 8 \}$, determine which of the following statements are true: (Choice A) $\{6, 7\} \subset A$

1. The set is defined as $$A = \{ \{\text{x, 1}\}, \{\text{y, 2}\}, \{\text{z, 3}\} \}.$$ 2. To evaluate the truth of each statement, let's analyze each carefully: