Question

Suppose ∣S∣ = 3 and ∣S×T∣ = 12. What is the cardinality of T ?

a) ∣T∣=4
b) ∣T∣=1
c) ∣T∣=36
d) ∣T∣=12
e) ∣T∣=7

Answer 1

Suppose ∣S∣ = 3 and ∣S×T∣ = 12. ∣T∣=4 is the cardinality of T. Thus the correct option is (a).

Step-by-step explanation:

The cardinality of the Cartesian product of two sets, denoted as ∣S×T∣, is equal to the product of the cardinalities of the individual sets S and T. Mathematically, ∣S×T∣ = ∣S∣ * ∣T∣.

Given that ∣S∣ = 3 and ∣S×T∣ = 12, we can use this information to find the cardinality of set T. Rearranging the formula, we have ∣T∣ = ∣S×T∣ / ∣S∣. Substituting the given values, we get ∣T∣ = 12 / 3 = 4.

Therefore, the correct answer is ∣T∣=4, which corresponds to option (a).

In summary, the cardinality of set T is determined by the given relationship between the cardinality of the Cartesian product of sets S and T. By applying the formula and substituting the known values, we find that ∣T∣ is indeed equal to 4. This understanding is crucial for solving problems involving set cardinalities and Cartesian products, providing a clear and concise solution to the given question.