10 maths chapter 1 Exercise 1.1

Exercise 1.1 Solutions

1. Find \( A \times B \), \( A \times A \) and \( B \times A \):

(i) \( A = \{2, -2, 3\} \) and \( B = \{1, -4\} \)

Given: \( A = \{2, -2, 3\} \), \( B = \{1, -4\} \)

\( A \times B \): \( \{(2, 1), (2, -4), (-2, 1), (-2, -4), (3, 1), (3, -4)\} \)

\( A \times A \): \( \{(2, 2), (2, -2), (2, 3), (-2, 2), (-2, -2), (-2, 3), (3, 2), (3, -2), (3, 3)\} \)

\( B \times A \): \( \{(1, 2), (1, -2), (1, 3), (-4, 2), (-4, -2), (-4, 3)\} \)

(ii) \( A = B = \{p, q\} \)

Here \( A = \{p, q\} \) and \( B = \{p, q\} \)

Since \( A = B \), \( A \times B = A \times A = B \times A \):

Result: \( \{(p, p), (p, q), (q, p), (q, q)\} \)

(iii) \( A = \{m, n\} \); \( B = \phi \)

\( A \times B \): \( \phi \) (The Cartesian product with an empty set is empty)

\( A \times A \): \( \{(m, m), (m, n), (n, m), (n, n)\} \)

\( B \times A \): \( \phi \)

2. Let \( A = \{1, 2, 3\} \) and \( B = \{x \mid x \text{ is a prime number less than } 10\} \). Find \( A \times B \) and \( B \times A \).

Given:

\( A = \{1, 2, 3\} \)

Prime numbers less than 10 are \( 2, 3, 5, 7 \). So, \( B = \{2, 3, 5, 7\} \)

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\( A \times B \):

\( \{(1, 2), (1, 3), (1, 5), (1, 7), (2, 2), (2, 3), (2, 5), (2, 7), (3, 2), (3, 3), (3, 5), (3, 7)\} \)

\( B \times A \):

\( \{(2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3), (5, 1), (5, 2), (5, 3), (7, 1), (7, 2), (7, 3)\} \)

3. If \( B \times A = \{(-2, 3), (-2, 4), (0, 3), (0, 4), (3, 3), (3, 4)\} \), find \( A \) and \( B \).

\( B = \text{Set of all first coordinates of elements of } B \times A = \{-2, 0, 3\} \)

\( A = \text{Set of all second coordinates of elements of } B \times A = \{3, 4\} \)

Hence: \( A = \{3, 4\} \) and \( B = \{-2, 0, 3\} \)

4. If \( A = \{5, 6\} \), \( B = \{4, 5, 6\} \), \( C = \{5, 6, 7\} \), Show that \( A \times A = (B \times B) \cap (C \times C) \).

LHS: \( A \times A \)

\( A \times A = \{(5, 5), (5, 6), (6, 5), (6, 6)\} \)   --- (1)

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RHS: \( (B \times B) \cap (C \times C) \)

\( B \times B = \{(4, 4), (4, 5), (4, 6), (5, 4), (5, 5), (5, 6), (6, 4), (6, 5), (6, 6)\} \)

\( C \times C = \{(5, 5), (5, 6), (5, 7), (6, 5), (6, 6), (6, 7), (7, 5), (7, 6), (7, 7)\} \)

Common elements of \( B \times B \) and \( C \times C \):

\( (B \times B) \cap (C \times C) = \{(5, 5), (5, 6), (6, 5), (6, 6)\} \)   --- (2)

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From (1) and (2), LHS = RHS. Hence Proved.

5. Given \( A=\{1, 2, 3\} \), \( B=\{2, 3, 5\} \), \( C=\{3, 4\} \), and \( D=\{1, 3, 5\} \), check if \( (A \cap C) \times (B \cap D) = (A \times B) \cap (C \times D) \) is true?

LHS: \( (A \cap C) \times (B \cap D) \)

\( A \cap C = \{1, 2, 3\} \cap \{3, 4\} = \{3\} \)

\( B \cap D = \{2, 3, 5\} \cap \{1, 3, 5\} = \{3, 5\} \)

\( (A \cap C) \times (B \cap D) = \{3\} \times \{3, 5\} = \{(3, 3), (3, 5)\} \)   --- (1)

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RHS: \( (A \times B) \cap (C \times D) \)

\( A \times B = \{(1,2), (1,3), (1,5), (2,2), (2,3), (2,5), (3,2), (3,3), (3,5)\} \)

\( C \times D = \{(3,1), (3,3), (3,5), (4,1), (4,3), (4,5)\} \)

\( (A \times B) \cap (C \times D) = \{(3, 3), (3, 5)\} \)   --- (2)

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From (1) and (2), LHS = RHS. Yes, it is true.

6. Let \( A = \{x \in W \mid x < 2\} \), \( B = \{x \in N \mid 1 < x \le 4\} \) and \( C = \{3, 5\} \).

First, write sets in roster form:

\( A = \{0, 1\} \)  (Whole numbers less than 2)

\( B = \{2, 3, 4\} \)  (Natural numbers greater than 1 and less than or equal to 4)

\( C = \{3, 5\} \)

(i) Verify: \( A \times (B \cup C) = (A \times B) \cup (A \times C) \)

\( B \cup C = \{2, 3, 4, 5\} \)

LHS = \( A \times (B \cup C) = \{(0,2), (0,3), (0,4), (0,5), (1,2), (1,3), (1,4), (1,5)\} \)

\( A \times B = \{(0,2), (0,3), (0,4), (1,2), (1,3), (1,4)\} \)

\( A \times C = \{(0,3), (0,5), (1,3), (1,5)\} \)

RHS = \( (A \times B) \cup (A \times C) = \{(0,2), (0,3), (0,4), (0,5), (1,2), (1,3), (1,4), (1,5)\} \)

LHS = RHS (Verified)

(ii) Verify: \( A \times (B \cap C) = (A \times B) \cap (A \times C) \)

\( B \cap C = \{3\} \)

LHS = \( A \times (B \cap C) = \{(0,3), (1,3)\} \)

RHS = \( (A \times B) \cap (A \times C) = \{(0,3), (1,3)\} \)

LHS = RHS (Verified)

(iii) Verify: \( (A \cup B) \times C = (A \times C) \cup (B \times C) \)

\( A \cup B = \{0, 1, 2, 3, 4\} \)

LHS = \( (A \cup B) \times C = \{(0,3), (0,5), (1,3), (1,5), (2,3), (2,5), (3,3), (3,5), (4,3), (4,5)\} \)

\( A \times C = \{(0,3), (0,5), (1,3), (1,5)\} \)

\( B \times C = \{(2,3), (2,5), (3,3), (3,5), (4,3), (4,5)\} \)

RHS = \( (A \times C) \cup (B \times C) = \{(0,3), (0,5), (1,3), (1,5), (2,3), (2,5), (3,3), (3,5), (4,3), (4,5)\} \)

LHS = RHS (Verified)

7. Let \( A = \) Natural numbers less than 8, \( B = \) Prime numbers less than 8, \( C = \) Set of even prime number.

Roster form of sets:

\( A = \{1, 2, 3, 4, 5, 6, 7\} \)

\( B = \{2, 3, 5, 7\} \)

\( C = \{2\} \)

(i) Verify: \( (A \cap B) \times C = (A \times C) \cap (B \times C) \)

\( A \cap B = \{2, 3, 5, 7\} \)

LHS = \( (A \cap B) \times C = \{(2, 2), (3, 2), (5, 2), (7, 2)\} \)

\( A \times C = \{(1,2), (2,2), (3,2), (4,2), (5,2), (6,2), (7,2)\} \)

\( B \times C = \{(2,2), (3,2), (5,2), (7,2)\} \)

RHS = \( (A \times C) \cap (B \times C) = \{(2, 2), (3, 2), (5, 2), (7, 2)\} \)

LHS = RHS (Verified)

(ii) Verify: \( A \times (B - C) = (A \times B) - (A \times C) \)

\( B - C = \{2, 3, 5, 7\} - \{2\} = \{3, 5, 7\} \)

LHS = \( A \times (B - C) = \{1, 2, 3, 4, 5, 6, 7\} \times \{3, 5, 7\} \)

LHS = \( \{(1,3),(1,5),(1,7),(2,3),(2,5),(2,7),(3,3),(3,5),(3,7),(4,3),(4,5),(4,7),(5,3),(5,5),(5,7),(6,3),(6,5),(6,7),(7,3),(7,5),(7,7)\} \)

RHS = \( (A \times B) - (A \times C) \)

Remove elements of \( A \times C \) (elements with second coordinate 2) from \( A \times B \).

RHS = LHS. Verified.