Question

For the sets A = {a, b} and B = {1, 2, 3}, select the false statement.

a. |A×B|=5
b. (b,3)∈A×B
c. A∩A²=∅
d. (b,a)∈A²

Answer 1

The false statement is option a: |A×B|=5, because the correct size of the Cartesian product of sets A and B is 6, not 5.

Step-by-step explanation:

Among the given options related to the sets A = {a, b} and B = {1, 2, 3}, we need to identify the false statement.

• |A×B| refers to the cardinality of the Cartesian product of sets A and B. The correct cardinality of A×B in this case would be 6, because there are 2 elements in A and 3 elements in B, giving us a total of 2*3=6 ordered pairs.
• (b,3) ∈ A×B is an ordered pair indicating that b is from set A and 3 is from set B; this is a true statement.
• A∩A² refers to the intersection of set A with its Cartesian product with itself, which should not be an empty set because (a,a) and (b,b) would be elements of A². Therefore, their intersection cannot be the empty set.
• (b,a) ∈ A² indicates that the ordered pair (b,a) is in the Cartesian product of set A with itself, which is a true statement since both elements are from set A.

Thus, the false statement is option a: |A×B|=5. The correct calculation of the Cartesian product's cardinality would indicate that the size of A×B is indeed 6, not 5.