State whether each of the following statements are true or false. If the statement is false, rewrite the given statement correctly. (i) If $P = \{m, n\}$ and $Q = \{n, m\}$, then $P imes Q = \{(m, n), (n, m)\}$. (ii) If $A$ and $B$ are non-empty sets, then $A imes B$ is a non

Question

State whether each of the following statements are true or false. If the statement is false, rewrite the given statement correctly. (i) If $P = \{m, n\}$ and $Q = \{n, m\}$, then $P 	imes Q = \{(m, n), (n, m)\}$. (ii) If $A$ and $B$ are non-empty sets, then $A 	imes B$ is a non-empty set of ordered pairs $(x, y)$ such that $x \in A$ and $y \in B$. (iii) If $A = \{1, 2\}$, $B = \{3, 4\}$, then $A 	imes (B \cap \phi) = \phi$.

Accepted Answer

> [!TLDR]
> The Cartesian product $P 	imes Q$ is the set of **all** ordered pairs from $P$ to $Q$, not just two. Statement (i) is false because it omits $(m,m)$ and $(n,n)$. Statement (ii) is true by definition. Statement (iii) is true because $B \cap \phi = \phi$, and any Cartesian product with an empty set is empty.

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### The Core Idea: What a Cartesian Product Really Is

The Cartesian product $A 	imes B$ is not about picking one element from $A$ and one from $B$ in some clever order. It is about **every possible pairing** — the first coordinate from $A$, the second from $B$, without skipping any combination. If $A$ has $p$ elements and $B$ has $q$ elements, then $A 	imes B$ has exactly $p 	imes q$ ordered pairs. That multiplication is the whole point of the name.

Now let’s test each statement against this definition.

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### 1. Statement (i): $P = \{m, n\}$, $Q = \{n, m\}$, then $P 	imes Q = \{(m, n), (n, m)\}$

First, note that $P$ and $Q$ are actually the same set: $\{m, n\}$. Order inside a set doesn’t matter, so $P = Q = \{m, n\}$.

The Cartesian product $P 	imes Q$ must contain **all** ordered pairs where the first entry is from $P$ and the second from $Q$. Since both sets have two elements, there are $2 	imes 2 = 4$ pairs:

- First element $m$: $(m, m)$ and $(m, n)$
- First element $n$: $(n, m)$ and $(n, n)$

So the full product is:
$$
P 	imes Q = \{(m, m), (m, n), (n, m), (n, n)\}
$$

The given statement lists only two pairs: $(m, n)$ and $(n, m)$. It misses $(m, m)$ and $(n, n)$. Therefore the statement is **false**.

> [!WARNING]
> A common mistake is to think that because $P$ and $Q$ contain the same elements, the product only contains pairs with different elements. That is not correct — the definition never excludes pairs where the coordinates are equal. Every combination counts.

**Corrected statement:** If $P = \{m, n\}$ and $Q = \{n, m\}$, then $P 	imes Q = \{(m, m), (m, n), (n, m), (n, n)\}$.

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### 2. Statement (ii): If $A$ and

Q.State whether each of the following statements are true or false. If the statement is false, rewrite the given statement correctly.

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

The Core Idea: What a Cartesian Product Really Is

1. Statement (i): $P = \{m, n\}$ , $Q = \{n, m\}$ , then $P \times Q = \{(m, n), (n, m)\}$

2. Statement (ii): If $A$ and $B$ are non-empty sets, then $A \times B$ is a non-empty set of ordered pairs $(x, y)$ such that $x \in A$ and $y \in B$ . …

Q.State whether each of the following statements are true or false. If the statement is false, rewrite the given statement correctly.

The Core Idea: What a Cartesian Product Really Is

1. Statement (i): P={m,n}P = \{m, n\}P={m,n}, Q={n,m}Q = \{n, m\}Q={n,m}, then P×Q={(m,n),(n,m)}P \times Q = \{(m, n), (n, m)\}P×Q={(m,n),(n,m)}

2. Statement (ii): If AAA and BBB are non-empty sets, then A×BA \times BA×B is a non-empty set of ordered pairs (x,y)(x, y)(x,y) such that x∈Ax \in Ax∈A and y∈By \in By∈B. …

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1. Statement (i): $P = \{m, n\}$ , $Q = \{n, m\}$ , then $P \times Q = \{(m, n), (n, m)\}$

2. Statement (ii): If $A$ and $B$ are non-empty sets, then $A \times B$ is a non-empty set of ordered pairs $(x, y)$ such that $x \in A$ and $y \in B$ . …

Unlock everything free for 14 days