Obtain the Cartesian product of sets A and B if A = {2, 3, 4} and B = {5, 6, 7}.
Understand the Problem
The question asks to find the Cartesian product of two sets A and B, where A = {2, 3, 4} and B = {5, 6, 7}. The Cartesian product is the set of all ordered pairs (a, b) where a is in A and b is in B.
Answer
$A \times B = \{(2, 5), (2, 6), (2, 7), (3, 5), (3, 6), (3, 7), (4, 5), (4, 6), (4, 7)\}$
Answer for screen readers
$A \times B = {(2, 5), (2, 6), (2, 7), (3, 5), (3, 6), (3, 7), (4, 5), (4, 6), (4, 7)}$
Steps to Solve
- Form ordered pairs with 2 from A and each element from B
We pair the first element of set A, which is 2, with each element of set B: 5, 6, and 7. This gives us the ordered pairs (2, 5), (2, 6), and (2, 7).
- Form ordered pairs with 3 from A and each element from B
Next, we pair the second element of set A, which is 3, with each element of set B: 5, 6, and 7. This gives us the ordered pairs (3, 5), (3, 6), and (3, 7).
- Form ordered pairs with 4 from A and each element from B
Then, we pair the third element of set A, which is 4, with each element of set B: 5, 6, and 7. This gives us the ordered pairs (4, 5), (4, 6), and (4, 7).
- Combine all ordered pairs into a set
Finally, we gather all the ordered pairs created in the previous steps into a set, which represents the Cartesian product $A \times B$.
$$ A \times B = {(2, 5), (2, 6), (2, 7), (3, 5), (3, 6), (3, 7), (4, 5), (4, 6), (4, 7)} $$
$A \times B = {(2, 5), (2, 6), (2, 7), (3, 5), (3, 6), (3, 7), (4, 5), (4, 6), (4, 7)}$
More Information
The Cartesian product of two sets creates a new set of ordered pairs, showing every possible combination of elements from the original sets. The number of elements in the Cartesian product is the product of the number of elements in each of the original sets. In this case, set A has 3 elements and set B has 3 elements, so their Cartesian product has $3 \times 3 = 9$ elements.
Tips
A common mistake is to mix up the order of elements in the ordered pairs i.e writing (5,2) instead of (2,5), or omitting some of the pairs, or including incorrect pairs.
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