answer:To prove that the number of elements in P(A) is greater than the number of elements in A, we will first find the elements of P(A) and then compare the number of elements in both sets. A = {1, 2, 3} The power set P(A) is the set of all possible subsets of A, including the empty set and A itself. So, let's list all the subsets of A: 1. {} (the empty set) 2. {1} 3. {2} 4. {3} 5. {1, 2} 6. {1, 3} 7. {2, 3} 8. {1, 2, 3} Now, let's count the number of elements in both sets: |A| = 3 (since A has three elements: 1, 2, and 3) |P(A)| = 8 (since P(A) has eight elements, as listed above) Since |P(A)| = 8 and |A| = 3, we can conclude that the number of elements in P(A) is greater than the number of elements in A. In general, for a set A with n elements, the number of elements in its power set P(A) is 2^n. In this case, since |A| = 3, |P(A)| = 2^3 = 8., The power set of a set A, denoted by P(A), is the set of all possible subsets of A. In this case, let A = {1, 2, 3}. We need to find the number of elements in both A and P(A) and compare them. First, let's find the number of elements in A. Since A contains the distinct elements 1, 2, and 3, we can simply count them to determine that |A| = 3. Next, let's find the number of elements in P(A). To do this, we need to list down all the possible subsets of A: {} (the empty set) {1} {2} {3} {1, 2} {1, 3} {2, 3} {1, 2, 3} We have found 8 distinct subsets of A. Therefore, |P(A)| = 8. Since 8 (the number of elements in P(A)) is greater than 3 (the number of elements in A), we have proven that the number of elements in the power set P(A) is indeed greater than the number of elements in the original set A.,To prove that the number of elements in the power set P(A) is greater than the number of elements in A, we can simply compute and compare both sets. The set A has 3 elements: {1, 2, 3}. Therefore, the number of elements in A is 3. To find the power set P(A), we take all possible subsets of A, including the empty set and A itself. These subsets are: 1. {} (empty set) 2. {1} 3. {2} 4. {3} 5. {1, 2} 6. {1, 3} 7. {2, 3} 8. {1, 2, 3} Therefore, the power set P(A) has 8 elements: { {}, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3} }. Comparing the two sets, the number of elements in P(A) (8) is indeed greater than the number of elements in A (3). Thus, we have proven that |P(A)| > |A|.

answer:A x B and B x A are two different Cartesian products, which means they are sets of ordered pairs where the first element comes from set A and the second element comes from set B for A x B, and the roles are reversed for B x A. A x B = {(1, 4), (1, 5), (2, 4), (2, 5), (3, 4), (3, 5)} B x A = {(4, 1), (4, 2), (4, 3), (5, 1), (5, 2), (5, 3)} Now let's compare the two sets: A x B contains ordered pairs with the first element from set A and the second element from set B, while B x A contains ordered pairs with the first element from set B and the second element from set A. For example, (1, 4) is in A x B but not in B x A, and (4, 1) is in B x A but not in A x B. Therefore, A x B is not equal to B x A, as they are two different sets of ordered pairs.,A x B is the Cartesian product of sets A and B, which consists of all ordered pairs (a, b) where a is an element of A and b is an element of B. A x B = {(1, 4), (1, 5), (2, 4), (2, 5), (3, 4), (3, 5)} B x A is the Cartesian product of sets B and A, which consists of all ordered pairs (b, a) where b is an element of B and a is an element of A. B x A = {(4, 1), (4, 2), (4, 3), (5, 1), (5, 2), (5, 3)} Now, let's compare A x B and B x A: A x B = {(1, 4), (1, 5), (2, 4), (2, 5), (3, 4), (3, 5)} B x A = {(4, 1), (4, 2), (4, 3), (5, 1), (5, 2), (5, 3)} As we can see, the ordered pairs in A x B are not the same as the ordered pairs in B x A. For example, (1, 4) is in A x B but not in B x A, and (4, 1) is in B x A but not in A x B. Therefore, A x B is not equal to B x A., To find the Cartesian product of A and B (denoted as A x B), we need to find all possible ordered pairs (a, b) where a is an element of set A and b is an element of set B. A x B = {(1, 4), (1, 5), (2, 4), (2, 5), (3, 4), (3, 5)} Now let's find B x A: B x A = {(4, 1), (4, 2), (4, 3), (5, 1), (5, 2), (5, 3)} We can see that A x B is not equal to B x A because they contain different ordered pairs even though they have the same number of elements. Specifically, the ordered pair (1, 4) is in A x B but not in B x A, and the ordered pair (4, 1) is in B x A but not in A x B.

answer:To find the Cartesian product of two sets, we multiply each element of the first set by each element of the second set. A × B = {(a, b) | a ∈ A and b ∈ B} A × B = {(1, 3), (1, 4), (2, 3), (2, 4)} B × A = {(b, a) | b ∈ B and a ∈ A} B × A = {(3, 1), (3, 2), (4, 1), (4, 2)} Since the ordered pairs in A × B and B × A are not the same, A × B is not equal to B × A.,To find A × B and B × A, I need to find all possible ordered pairs formed by the elements of A and B. A × B = {(a, b) | a ∈ A, b ∈ B} A × B = {(1, 3), (1, 4), (2, 3), (2, 4)} For B × A, I will repeat the same process but this time with the elements of B as the first element in each ordered pair. B × A = {(b, a) | b ∈ B, a ∈ A} B × A = {(3, 1), (3, 2), (4, 1), (4, 2)} Now, let's compare A × B and B × A. As you can see, the ordered pairs in each set are different, except for the fourth ordered pair in both cases, which is (2, 4). The ordered pairs (1, 3), (1, 4), (2, 3), (3, 1), and (3, 2) are not in the other set, which shows that A × B is not equal to B × A.,A × B represents the Cartesian product of sets A and B. The Cartesian product of two sets is the set of all ordered pairs (a, b) where a is an element of A and b is an element of B. A × B = {(a, b) | a ∈ A, b ∈ B} So, for A = {1, 2} and B = {3, 4}, we have: A × B = {(1, 3), (1, 4), (2, 3), (2, 4)} Now, let's find B × A: B × A = {(b, a) | b ∈ B, a ∈ A} B × A = {(3, 1), (3, 2), (4, 1), (4, 2)} Comparing the two sets A × B and B × A, we can see that they are not equal: A × B = {(1, 3), (1, 4), (2, 3), (2, 4)} B × A = {(3, 1), (3, 2), (4, 1), (4, 2)} The ordered pairs in A × B and B × A are different, so A × B is not equal to B × A.

answer:The Cartesian product A x B is defined as the set of all ordered pairs (a, b) where a is an element of set A and b is an element of set B. A x B = {(1, 4), (1, 5), (2, 4), (2, 5), (3, 4), (3, 5)} Similarly, the Cartesian product B x A is defined as the set of all ordered pairs (b, a) where b is an element of set B and a is an element of set A. B x A = {(4, 1), (4, 2), (4, 3), (5, 1), (5, 2), (5, 3)} We can see that A x B ≠ B x A, which shows that the Cartesian product of two sets is not commutative.,To find the Cartesian product of two sets, we take each element of the first set and pair it with each element of the second set. 1. Finding A x B: A x B = {(1, 4), (1, 5), (2, 4), (2, 5), (3, 4), (3, 5)} 2. Finding B x A: B x A = {(4, 1), (4, 2), (4, 3), (5, 1), (5, 2), (5, 3)} Now, let's compare the two Cartesian products: A x B ≠ B x A This shows that the Cartesian product is not commutative because the order of the elements in the resulting pairs is different when switching the sets (A and B) in the Cartesian product operation.,The Cartesian product of two sets A and B, denoted as A x B, is the set of all ordered pairs (a, b) where a is an element of A and b is an element of B. Let's find A x B and B x A: A x B = {(1, 4), (1, 5), (2, 4), (2, 5), (3, 4), (3, 5)} B x A = {(4, 1), (4, 2), (4, 3), (5, 1), (5, 2), (5, 3)} Now, let's compare the two sets: A x B ≠ B x A {(1, 4), (1, 5), (2, 4), (2, 5), (3, 4), (3, 5)} ≠ {(4, 1), (4, 2), (4, 3), (5, 1), (5, 2), (5, 3)} As we can see, the Cartesian product of two sets is not commutative, since the ordered pairs in A x B are not the same as the ordered pairs in B x A.