Let $A$ and $B$ be two sets such that $n(A) = 3$ and $n(B) = 2$. If $(x, 1)$, $(y, 2)$, $(z, 1)$ are in $A imes B$, find $A$ and $B$, where $x$, $y$ and $z$ are distinct elements.

Question

Let $A$ and $B$ be two sets such that $n(A) = 3$ and $n(B) = 2$. If $(x, 1)$, $(y, 2)$, $(z, 1)$ are in $A 	imes B$, find $A$ and $B$, where $x$, $y$ and $z$ are distinct elements.

Accepted Answer

> [!TLDR]
> The Cartesian product $A 	imes B$ consists of all ordered pairs $(a, b)$ where $a \in A$ and $b \in B$. Since the second coordinates of the given pairs tell us what's in $B$, and the first coordinates tell us what's in $A$, we find $A = \{x, y, z\}$ and $B = \{1, 2\}$.

The Cartesian product $A 	imes B$ is the set of all ordered pairs where the first element comes from $A$ and the second from $B$. This means every ordered pair $(a, b) \in A 	imes B$ satisfies $a \in A$ and $b \in B$.

When we're told that certain pairs belong to $A 	imes B$, we can work backwards: the first coordinates must be elements of $A$, and the second coordinates must be elements of $B$. This reverse-engineering is the key to finding the sets.

**Finding set $B$:**

1. Look at the second coordinates of all given pairs: $(x, 1)$, $(y, 2)$, $(z, 1)$.
   
   The second coordinates are $1$, $2$, and $1$. Since these pairs are in $A 	imes B$, each second coordinate must be an element of $B$.

2. The distinct second coordinates are $1$ and $2$, so we know $\{1, 2\} \subseteq B$.

3. We're told $n(B) = 2$, meaning $B$ has exactly two elements. Since we've already identified two elements that must be in $B$, we have $B = \{1, 2\}$.

**Finding set $A$:**

4. Now examine the first coordinates: $x$, $y$, and $z$. Since each pair is in $A 	imes B$, we know $x \in A$, $y \in A$, and $z \in A$.

5. The problem states that $x$, $y$, and $z$ are distinct elements, so $\{x, y, z\} \subseteq A$ and these are three different elements. …

Q.Let $A$ and $B$ be two sets such that $n(A) = 3$ and $n(B) = 2$ . If $(x, 1)$ , $(y, 2)$ , $(z, 1)$ are in $A \times B$ , find $A$ and $B$ , where $x$ , $y$ and $z$ are distinct elements.

Concept understanding — Cartesian Product

Cartesian Product: From Intuition to Definition

The Intuition

The Precise Definition

Key Properties to Remember

Examples to Cement the Idea

Why This Matters

Q.Let AAA and BBB be two sets such that n(A)=3n(A) = 3n(A)=3 and n(B)=2n(B) = 2n(B)=2. If (x,1)(x, 1)(x,1), (y,2)(y, 2)(y,2), (z,1)(z, 1)(z,1) are in A×BA \times BA×B, find AAA and BBB, where xxx, yyy and zzz are distinct elements.

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Q.Let $A$ and $B$ be two sets such that $n(A) = 3$ and $n(B) = 2$ . If $(x, 1)$ , $(y, 2)$ , $(z, 1)$ are in $A \times B$ , find $A$ and $B$ , where $x$ , $y$ and $z$ are distinct elements.

Unlock everything free for 14 days