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Q4(i):
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 $P \times Q = \{(m, n),(n, m)\}$.
Solution :
Step 1: Identifying the Cartesian Product Definition
The Cartesian product of two non-empty sets $P$ and $Q$, denoted as $P \times Q$, is defined as the set of all ordered pairs $(p, q)$ such that $p \in P$ and $q \in Q$. Mathematically: $P \times Q = \{ (p, q) : p \in P, q \in Q \}$ [Per the definition of Cartesian Product in Set Theory].
Step 2: Analysis of Given Sets
Given sets are:
- $P = \{m, n\}$
- $Q = \{n, m\}$
The number of elements in the Cartesian product $P \times Q$ must be equal to the product of the cardinalities of the individual sets. Given $n(P) = 2$ and $n(Q) = 2$, the number of elements in $P \times Q$ is $2 \times 2 = 4$ [Per the Multiplication Principle of Sets].
Step 3: Calculating the Cartesian Product $P \times Q$
To find $P \times Q$, we take each element of $P$ and pair it with every element of $Q$:
- For $p = m$: $(m, n), (m, m)$
- For $p = n$: $(n, n), (n, m)$
Therefore, $P \times Q = \{ (m, n), (m, m), (n, n), (n, m) \}$.
Step 4: Evaluating the Truth Value
The original statement claims $P \times Q = \{(m, n), (n, m)\}$. Since our calculated set contains 4 elements and the statement suggests only 2, the statement is False.
Step 5: Rewriting the Correct Statement
The statement is false because the Cartesian product of two sets containing 2 elements each must contain $2^2 = 4$ ordered pairs.
Correct Statement: If $P = \{m, n\}$ and $Q = \{n, m\}$, then $P \times Q = \{(m, n), (m, m), (n, n), (n, m)\}$.
Final Solution: The statement is False. The correct statement is: If $P = \{m, n\}$ and $Q = \{n, m\}$, then $P \times Q = \{(m, n), (m, m), (n, n), (n, m)\}$.
More Questions from Class 11 Mathematics Relations and Functions EXERCISE 2.1
- Q1: If $(\frac{x}{3} + 1, y - \frac{2}{3}) = (\frac{5}{3}, \frac{1}{3})$, find the values of $x$ and $y$.
- Q2: If the set $A$ has 3 elements and the set $B = \{3, 4, 5\}$, then find the number of elements in $(A \times B)$.
- Q3: If $G = \{7, 8\}$ and $H = \{5, 4, 2\}$, find $G \times H$ and $H \times G$.
- Q4(ii): State whether each of the following statements are true or false. If the statement is false, rewrite the given statement correctly. (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 $y \in B$.
- Q4(iii): State whether each of the following statements are true or false. If the statement is false, rewrite the given statement correctly. (iii) If $A = \{1, 2\}$, $B = \{3, 4\}$, then $A \times (B \cap \phi) = \phi$.
- Q5: If $A = \{–1, 1\}$, find $A \times A \times A$.
- Q6: If $A \times B = \{(a, x),(a, y), (b, x), (b, y)\}$. Find $A$ and $B$.
- Q7(i): Let $A = \{1, 2\}$, $B = \{1, 2, 3, 4\}$, $C = \{5, 6\}$ and $D = \{5, 6, 7, 8\}$. Verify that (i) $A \times (B \cap C) = (A \times B) \cap (A \times C)$.
- Q7(ii): Let $A = \{1, 2\}$, $B = \{1, 2, 3, 4\}$, $C = \{5, 6\}$ and $D = \{5, 6, 7, 8\}$. Verify that (ii) $A \times C$ is a subset of $B \times D$.
- Q8: Let $A = \{1, 2\}$ and $B = \{3, 4\}$. Write $A \times B$. How many subsets will $A \times B$ have? List them.
- Q9: 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.
- Q10: The Cartesian product $A \times A$ has 9 elements among which are found $(–1, 0)$ and $(0,1)$. Find the set $A$ and the remaining elements of $A \times A$.
CBSE Solutions for Class 11 Mathematics Relations and Functions
Chapters in CBSE - Class 11 Mathematics
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