Understanding the Problem

We are given two sets: A with 2 elements and B with 4 elements. The Cartesian product A × B has 2 × 4 = 8 elements. We need to find the number of subsets of A × B that have 3 or more elements.

Step 1: Find Total Number of Subsets of A × B

Since A × B has 8 elements, the total number of subsets is 2⁸ = 256. This includes the empty set and all subsets of various sizes.

Step 2: Find Number of Subsets with Fewer than 3 Elements

We subtract the number of subsets with 0, 1, or 2 elements from the total to get subsets with 3 or more elements.

Number of subsets with 0 elements (empty set): $C(8,0)=1$

Number of subsets with 1 element: $C(8,1)=8$

Number of subsets with 2 elements: $C(8,2)=\frac{8\times 7}{2\times 1}=28$

Total subsets with fewer than 3 elements: 1 + 8 + 28 = 37

Step 3: Find Number of Subsets with 3 or More Elements

Subtract the number from Step 2 from the total subsets:

$256-37=219$

Final Answer

The number of subsets of A × B having 3 or more elements is 219.

Related Topics

• Set Theory
• Cartesian Product
• Combinatorics
• Subsets and Power Set

Key Formulae

Number of elements in Cartesian product A × B: |A| × |B|

Number of subsets of a set with n elements: 2ⁿ

Number of ways to choose k elements from n: $C(n,k)=\frac{n!}{k!(n-k)!}$