In the questions below determine the size (i.e. cardinality)
of the set.
• {x|x ∈ N, x2 < 10}
• P({a,b,c,d,e }) where P denotes the Power set
• A × B where A = {1, 2, 3, 4, 5} and B = {1, 2, 3}
• {x|x ∈ Z, 9x2 −1 = 0}
• A × B where A = {a, b, c} and B = ∅
To determine the size (cardinality) of a set, you need to count the number of elements in the set. Here's how you can find the size of each set:
1. {x|x ∈ N, x^2 < 10}:
- N represents the set of natural numbers (positive integers starting from 1).
- To find the elements of the set, you need to consider values of x that satisfy the condition x^2 < 10.
- The numbers that satisfy this condition are x = 1, 2, and 3, because 1^2 = 1, 2^2 = 4, and 3^2 = 9.
- Therefore, the size of the set is 3.
2. P({a,b,c,d,e}):
- P denotes the power set, which is the set of all subsets of a given set.
- The given set is {a, b, c, d, e}, which has 5 elements.
- A power set includes all possible subsets, including the empty set and the set itself.
- The number of subsets in a power set can be calculated using 2^n, where n is the number of elements in the original set.
- In this case, we have 5 elements, so the size of the power set is 2^5 = 32.
3. A × B where A = {1, 2, 3, 4, 5} and B = {1, 2, 3}:
- A × B represents the Cartesian product of sets A and B.
- The Cartesian product is formed by pairing each element of set A with each element of set B.
- In this case, A has 5 elements and B has 3 elements.
- To find the size of the Cartesian product, simply multiply the number of elements in A by the number of elements in B.
- So, the size of the set A × B is 5 × 3 = 15.
4. {x|x ∈ Z, 9x^2 −1 = 0}:
- Z represents the set of integers (positive, negative, and zero).
- To find the elements of the set, you need to determine the values of x that satisfy the equation 9x^2 − 1 = 0.
- Solving this equation, you get 9x^2 = 1, where x^2 = 1/9.
- The solutions to this equation are x = 1/3 and x = -1/3.
- However, since the set only contains integers, the solution x = 1/3 is not included.
- Therefore, the size of the set is 1, with x = -1/3.
5. A × B where A = {a, b, c} and B = ∅:
- ∅ represents the empty set, which contains no elements.
- When you perform a Cartesian product between a set and the empty set, the resulting set will also be empty.
- Therefore, the size of the set A × B is 0.