Question

Determine whether each set is finite or infinite. If a set is finite, find the cardinality. Write your answer using the notation of set cardinality. (a) W={…,−3,−2,−1,0,1,2,3} (b) X={4,8,12,16,…,100} (c) Y={x∣x∈R and x∈/Q} (d) Z={x∣x∈Z and (x²−9)(x²+4)(2x+6)(3x−4)=0} Let A={5,6} and B={7,8}. (a) Find A×B (b) Find P(A×B)

Answer 1

The sets W, X, Y, and Z are determined to be finite or infinite, with their respective cardinalities. The cross product of sets A and B is found, along with the power set of A×B.

Step-by-step explanation:

(a) The set W = {..., -3, -2, -1, 0, 1, 2, 3} is finite. Its cardinality is 7, denoted as |W| = 7. This is because the set contains 7 distinct elements.

(b) The set X = {4, 8, 12, 16, ..., 100} is finite. Its cardinality is 25, denoted as |X| = 25. This is because the set contains 25 elements in a sequential pattern where each element is obtained by adding 4 to the previous element.

(c) The set Y = {x∣x∈R and x∈/Q} is infinite. This is because it represents the set of real numbers that are not rational, and the set of real numbers is infinite.

(d) The set Z = {x∣x∈Z and (x²−9)(x²+4)(2x+6)(3x−4)=0} is finite. Its cardinality is 4, denoted as |Z| = 4. This is because the equation will only have four solutions in the set of integers.

(a) A×B = {(5, 7), (5, 8), (6, 7), (6, 8)}. The set A×B contains all possible ordered pairs where the first element is from set A and the second element is from set B.

(b) P(A×B) = 4. The set A×B contains 4 possible outcomes.