State whether each of the following statements are true or false. If…

Question 4 of 33intermediate🔍 AnalyzeShort Answer2 marks

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 $P \times Q = \{(m, n), (n, m)\}$ .

Answer

False. Correct statement: If $P = \{m, n\}$ and $Q = \{n, m\}$ , then $P \times Q = \{(m, n), (m, m), (n, n), (n, m)\}$ .

(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 $y \in B$ .

Answer

True.

(iii)

If $A = \{1, 2\}$ , $B = \{3, 4\}$ , then $A \times (B \cap \phi) = \phi$ .

Answer

True.

Explanation

For part (i), the Cartesian product $P \times Q$ involves pairing every element of set $P$ with every element of set $Q$ . Since $P=\{m, n\}$ and $Q=\{n, m\}$ , there are $2 \times 2 = 4$ possible ordered pairs, not just the two listed. For part (ii), the statement correctly defines the Cartesian product for non-empty sets. For part (iii), the intersection of any set $B$ with the empty set $\phi$ is $\phi$ , and the Cartesian product of any set $A$ with an empty set results in an empty set.

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