Westonci.ca is your trusted source for finding answers to a wide range of questions, backed by a knowledgeable community. Explore thousands of questions and answers from a knowledgeable community of experts on our user-friendly platform. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.
Sagot :
Let's address both parts of the question step-by-step.
### Part i) Verification of [tex]\( A \times (B \cap C) = (A \times B) \cap (A \times C) \)[/tex]
1. Find [tex]\( B \cap C \)[/tex]:
[tex]\[ B = \{1, 2, 3, 4\} \][/tex]
[tex]\[ C = \{5, 6\} \][/tex]
[tex]\[ B \cap C = \{ \} \][/tex]
Since there are no common elements between [tex]\( B \)[/tex] and [tex]\( C \)[/tex], [tex]\( B \cap C \)[/tex] is the empty set, [tex]\( \{ \} \)[/tex].
2. Compute [tex]\( A \times (B \cap C) \)[/tex]:
[tex]\[ A \times \{ \} = \{ (a, b) \mid a \in A, b \in \{ \} \} = \{ \} \][/tex]
So, [tex]\( A \times (B \cap C) = \{ \} \)[/tex].
3. Find [tex]\( A \times B \)[/tex]:
[tex]\[ A = \{1, 2\} \][/tex]
[tex]\[ B = \{1, 2, 3, 4\} \][/tex]
[tex]\[ A \times B = \{ (1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (2, 4) \} \][/tex]
4. Find [tex]\( A \times C \)[/tex]:
[tex]\[ A = \{1, 2\} \][/tex]
[tex]\[ C = \{5, 6\} \][/tex]
[tex]\[ A \times C = \{ (1, 5), (1, 6), (2, 5), (2, 6) \} \][/tex]
5. Compute [tex]\( (A \times B) \cap (A \times C) \)[/tex]:
[tex]\[ (A \times B) \cap (A \times C) = \{ \} \][/tex]
Since there are no common ordered pairs between [tex]\( A \times B \)[/tex] and [tex]\( A \times C \)[/tex].
6. Verification:
[tex]\[ A \times (B \cap C) = \{ \} \][/tex]
[tex]\[ (A \times B) \cap (A \times C) = \{ \} \][/tex]
Therefore:
[tex]\[ A \times (B \cap C) = (A \times B) \cap (A \times C) \][/tex]
This verifies the first part.
### Part ii) Verification of [tex]\( A \times (B \cup C) = (A \times B) \cup (A \times C) \)[/tex]
1. Find [tex]\( B \cup C \)[/tex]:
[tex]\[ B = \{1, 2, 3, 4\} \][/tex]
[tex]\[ C = \{5, 6\} \][/tex]
[tex]\[ B \cup C = \{ 1, 2, 3, 4, 5, 6 \} \][/tex]
2. Compute [tex]\( A \times (B \cup C) \)[/tex]:
[tex]\[ A = \{1, 2\} \][/tex]
[tex]\[ B \cup C = \{ 1, 2, 3, 4, 5, 6 \} \][/tex]
[tex]\[ A \times (B \cup C) = \{ (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6) \} \][/tex]
3. Compute [tex]\( (A \times B) \cup (A \times C) \)[/tex]:
From the previous steps, we have:
[tex]\[ A \times B = \{ (1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (2, 4) \} \][/tex]
[tex]\[ A \times C = \{ (1, 5), (1, 6), (2, 5), (2, 6) \} \][/tex]
Combining these sets:
[tex]\[ (A \times B) \cup (A \times C) = \{ (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6) \} \][/tex]
4. Verification:
[tex]\[ A \times (B \cup C) = \{ (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6) \} \][/tex]
[tex]\[ (A \times B) \cup (A \times C) = \{ (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6) \} \][/tex]
Therefore:
[tex]\[ A \times (B \cup C) = (A \times B) \cup (A \times C) \][/tex]
This verifies the second part.
In conclusion, both identities have been verified as true.
### Part i) Verification of [tex]\( A \times (B \cap C) = (A \times B) \cap (A \times C) \)[/tex]
1. Find [tex]\( B \cap C \)[/tex]:
[tex]\[ B = \{1, 2, 3, 4\} \][/tex]
[tex]\[ C = \{5, 6\} \][/tex]
[tex]\[ B \cap C = \{ \} \][/tex]
Since there are no common elements between [tex]\( B \)[/tex] and [tex]\( C \)[/tex], [tex]\( B \cap C \)[/tex] is the empty set, [tex]\( \{ \} \)[/tex].
2. Compute [tex]\( A \times (B \cap C) \)[/tex]:
[tex]\[ A \times \{ \} = \{ (a, b) \mid a \in A, b \in \{ \} \} = \{ \} \][/tex]
So, [tex]\( A \times (B \cap C) = \{ \} \)[/tex].
3. Find [tex]\( A \times B \)[/tex]:
[tex]\[ A = \{1, 2\} \][/tex]
[tex]\[ B = \{1, 2, 3, 4\} \][/tex]
[tex]\[ A \times B = \{ (1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (2, 4) \} \][/tex]
4. Find [tex]\( A \times C \)[/tex]:
[tex]\[ A = \{1, 2\} \][/tex]
[tex]\[ C = \{5, 6\} \][/tex]
[tex]\[ A \times C = \{ (1, 5), (1, 6), (2, 5), (2, 6) \} \][/tex]
5. Compute [tex]\( (A \times B) \cap (A \times C) \)[/tex]:
[tex]\[ (A \times B) \cap (A \times C) = \{ \} \][/tex]
Since there are no common ordered pairs between [tex]\( A \times B \)[/tex] and [tex]\( A \times C \)[/tex].
6. Verification:
[tex]\[ A \times (B \cap C) = \{ \} \][/tex]
[tex]\[ (A \times B) \cap (A \times C) = \{ \} \][/tex]
Therefore:
[tex]\[ A \times (B \cap C) = (A \times B) \cap (A \times C) \][/tex]
This verifies the first part.
### Part ii) Verification of [tex]\( A \times (B \cup C) = (A \times B) \cup (A \times C) \)[/tex]
1. Find [tex]\( B \cup C \)[/tex]:
[tex]\[ B = \{1, 2, 3, 4\} \][/tex]
[tex]\[ C = \{5, 6\} \][/tex]
[tex]\[ B \cup C = \{ 1, 2, 3, 4, 5, 6 \} \][/tex]
2. Compute [tex]\( A \times (B \cup C) \)[/tex]:
[tex]\[ A = \{1, 2\} \][/tex]
[tex]\[ B \cup C = \{ 1, 2, 3, 4, 5, 6 \} \][/tex]
[tex]\[ A \times (B \cup C) = \{ (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6) \} \][/tex]
3. Compute [tex]\( (A \times B) \cup (A \times C) \)[/tex]:
From the previous steps, we have:
[tex]\[ A \times B = \{ (1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (2, 4) \} \][/tex]
[tex]\[ A \times C = \{ (1, 5), (1, 6), (2, 5), (2, 6) \} \][/tex]
Combining these sets:
[tex]\[ (A \times B) \cup (A \times C) = \{ (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6) \} \][/tex]
4. Verification:
[tex]\[ A \times (B \cup C) = \{ (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6) \} \][/tex]
[tex]\[ (A \times B) \cup (A \times C) = \{ (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6) \} \][/tex]
Therefore:
[tex]\[ A \times (B \cup C) = (A \times B) \cup (A \times C) \][/tex]
This verifies the second part.
In conclusion, both identities have been verified as true.
We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Thank you for visiting Westonci.ca. Stay informed by coming back for more detailed answers.
