Fractions and Decimals

Discrete Mathematics

Differential Equations

Solved on Jan 25, 2024

Find the Cartesian product of set A with the union of sets B and C, given $A=\{a, b\}, B=\{1,2\}$ , and $C=\{2,3\}$ .

STEP 1

Assumptions
1. Set $A$ contains elements $a$ and $b$ .
2. Set $B$ contains elements $1$ and $2$ .
3. Set $C$ contains elements $2$ and $3$ .
4. The Cartesian product $A \times (B \cup C)$ means all ordered pairs where the first element is from $A$ and the second element is from the union of $B$ and $C$ .

STEP 2

First, we need to find the union of sets $B$ and $C$ . The union of two sets includes all the distinct elements that are in either set.
$B \cup C = \{1, 2\} \cup \{2, 3\}$

STEP 3

Calculate the union of $B$ and $C$ by combining the elements from both sets, without repeating any elements.
$B \cup C = \{1, 2, 3\}$

STEP 4

Now, we need to find the Cartesian product of set $A$ and the union of sets $B$ and $C$ .
$A \times (B \cup C) = \{(x, y) | x \in A \text{ and } y \in B \cup C\}$

STEP 5

List all possible ordered pairs where the first element is from set $A$ and the second element is from the union $B \cup C$ .
$A \times (B \cup C) = \{(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3)\}$

STEP 6

Compare the result with the given options.
The correct answer is a. $A \times(B \cup C)=\{(a, 1),(a, 2),(a, 3),(b, 1),(b, 2),(b, 3)\}$ .

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