For i; j 2 N let (i; j) denote the ordered pair in the cartesian product N � N and fi; jg � N the
subset consisting of the elements i and j. De�ne the sets A and B as follows:
A := f;; 1; 2; (1; 3); (3; 1); f1; 3g; f3; 1gg, B := f3; (1; 2); (2; 1); f1; 2g; f2; 1; 2gg. Check whether the
following statements are true or false. Give a brief explanation for your answer.
(i): jAj = 6; (ii): jBj = 5; (iii): A \ B = ;; (iv): (3; 1) 2 A; (v): (1; 3) 2 A � B;
(vi): (3; 1) 2 A \ (B � A); (vii): (2; 1) 2 B \ (A � A); (viii): f1; 3g � A.
To answer these questions, we need to understand the definitions of sets A and B, as well as the operations involved.
First, let's define the sets A and B as given:
A := {∅, 1, 2, (1, 3), (3, 1), {1, 3}, {3, 1}}
B := {3, (1, 2), (2, 1), {1, 2}, {2, 1, 2}}
Now let's go through each statement and determine if it is true or false.
(i): jAj = 6
To find the size, or cardinality, of set A, we need to count the number of elements in it. Counting the elements, we can see that A contains 6 distinct elements: ∅, 1, 2, (1, 3), (3, 1), and {1, 3}. Therefore, jAj = 6. The statement is true.
(ii): jBj = 5
Similarly, we count the number of distinct elements in set B: 3, (1, 2), (2, 1), {1, 2}, and {2, 1, 2}. Therefore, jBj = 5. The statement is true.
(iii): A \ B = ∅
The set operation A \ B represents the set difference, which includes all the elements that are in A but not in B. Checking the elements in A that are not in B: ∅, 1, 2, (1, 3), and {1, 3}. None of these elements are in B. Therefore, A \ B = ∅. The statement is false.
(iv): (3, 1) ∈ A
To check if (3, 1) is an element of A, we look at the elements in A: ∅, 1, 2, (1, 3), (3, 1), {1, 3}, and {3, 1}. (3, 1) is present in A. Therefore, (3, 1) ∈ A. The statement is true.
(v): (1, 3) ∈ (A × B)
The expression A × B represents the Cartesian product of sets A and B, which includes all possible ordered pairs where the first element is from A and the second element is from B. Checking if (1, 3) is in A × B, we need to see if (1, 3) is a valid combination according to the definitions of A and B. Looking at A and B, we can see that (1, 3) is a valid combination. Therefore, (1, 3) ∈ (A × B). The statement is true.
(vi): (3, 1) ∈ A \ (B × A)
The expression B × A represents the Cartesian product of sets B and A. To determine if (3, 1) is in A \ (B × A), we need to check if it belongs to A but not in (B × A). Checking if (3, 1) is in (B × A), we need to examine the combinations of B and A. (3, 1) is not a valid combination according to the definitions of A and B. Therefore, (3, 1) ∉ (B × A). Checking if (3, 1) is in A, we can confirm that (3, 1) is indeed an element of A. Therefore, (3, 1) ∈ A \ (B × A). The statement is true.
(vii): (2, 1) ∈ B \ (A × A)
Again, A × A represents the Cartesian product of sets A and A. Checking if (2, 1) is in (A × A), we need to assess if it is a valid combination according to the definitions of A. Looking at A, we can see that (2, 1) is not a valid combination. Therefore, (2, 1) ∉ (A × A). Checking if (2, 1) is an element of B, we can confirm that (2, 1) is in B. Therefore, (2, 1) ∈ B \ (A × A). The statement is true.
(viii): {1, 3} ⊆ A
To check if {1, 3} is a subset of A, we need to verify if every element in {1, 3} is also in A. Checking the elements {1, 3} = {1, 3}, we can see that it is an element of A. Therefore, {1, 3} ⊆ A. The statement is true.
In summary:
(i): True
(ii): True
(iii): False
(iv): True
(v): True
(vi): True
(vii): True
(viii): True