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Q8:
Let $A = \{1, 2\}$ and $B = \{3, 4\}$. Write $A \times B$. How many subsets will $A \times B$ have? List them.
Solution :
Step 1: Cartesian Product Construction
The Cartesian product of two sets $A$ and $B$, denoted by $A \times B$, is the set of all ordered pairs $(a, b)$ such that $a \in A$ and $b \in B$. [Per the definition of Cartesian product in Set Theory].
Given:
- $A = \{1, 2\}$
- $B = \{3, 4\}$
Mapping elements of $A$ to $B$:
- $(1, 3)$
- $(1, 4)$
- $(2, 3)$
- $(2, 4)$
Thus, $A \times B = \{(1, 3), (1, 4), (2, 3), (2, 4)\}$.
Step 2: Calculating the Number of Subsets
The total number of subsets of any set $S$ is given by the formula $2^n$, where $n$ is the number of elements (cardinality) in the set $S$. [Per the Power Set Axiom].
Here, the cardinality of $A \times B$ is $|A \times B| = 4$.
Number of subsets $= 2^4 = 16$.
Step 3: Listing the Subsets
Let $S = A \times B = \{p_1, p_2, p_3, p_4\}$, where $p_1=(1,3), p_2=(1,4), p_3=(2,3), p_4=(2,4)$. The 16 subsets are listed as follows:
| Subset Size | Subsets |
|---|---|
| 0 (Empty Set) | $\emptyset$ |
| 1 | $\{(1, 3)\}, \{(1, 4)\}, \{(2, 3)\}, \{(2, 4)\}$ |
| 2 | $\{(1, 3), (1, 4)\}, \{(1, 3), (2, 3)\}, \{(1, 3), (2, 4)\}, \{(1, 4), (2, 3)\}, \{(1, 4), (2, 4)\}, \{(2, 3), (2, 4)\}$ |
| 3 | $\{(1, 3), (1, 4), (2, 3)\}, \{(1, 3), (1, 4), (2, 4)\}, \{(1, 3), (2, 3), (2, 4)\}, \{(1, 4), (2, 3), (2, 4)\}$ |
| 4 | $\{(1, 3), (1, 4), (2, 3), (2, 4)\}$ |
Visual Representation of $A \times B$
Final Solution: $A \times B = \{(1, 3), (1, 4), (2, 3), (2, 4)\}$. The set has $16$ subsets.
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(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 $P \times Q = \{(m, n),(n, m)\}$.
- 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 $B \times D$.
- 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 $A \times A$.
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