Set S has 4 elements ,A and B are subsets of S.the probability that A and B are not disjoint is

To find the probability that subsets A and B are not disjoint when set S has 4 elements, we start by understanding what it means for two sets to be disjoint. Two sets are disjoint if they have no elements in common. Therefore, the probability that A and B are *not* disjoint is equivalent to the probability that they do share at least one common element. Let's break this down step by step.

Understanding the Elements of Set S

Let’s denote the elements of set S as follows: S = {1, 2, 3, 4}. Since there are 4 elements in S, any subset A or B can have anywhere from 0 to 4 elements. This gives us a total of 2⁴ = 16 possible subsets for S (including the empty set).

Calculating the Total Possible Combinations

Each of subsets A and B can independently choose from any of the 16 subsets of S. Therefore, the total number of combinations of A and B together is:

• Total combinations = 16 (for A) × 16 (for B) = 256.

Finding the Disjoint Sets

Now, we need to determine how many combinations result in A and B being disjoint. For A and B to be disjoint, they cannot share any elements. For each element in S, there are three choices when forming disjoint sets:

• The element can be in subset A.
• The element can be in subset B.
• The element can be in neither A nor B.

This means that for each of the 4 elements in S, we have 3 independent choices. The total number of ways to choose A and B such that they are disjoint is:

• Disjoint combinations = 3⁴ = 81.

Calculating the Probability

Having established the total possible combinations and the combinations that result in disjoint sets, we can now calculate the probability that A and B are *not* disjoint. The number of combinations where A and B are not disjoint is simply the total combinations minus the disjoint combinations:

• Combinations where A and B are not disjoint = 256 - 81 = 175.

Now, the probability that A and B are not disjoint is calculated by dividing the number of combinations where they are not disjoint by the total number of combinations:

• Probability (A and B not disjoint) = 175 / 256.

Final Result

So, the probability that subsets A and B are not disjoint is:

• Probability = 175/256 ≈ 0.6836 or about 68.36%.

This means that there's a significant chance that A and B will share at least one element when selected from set S with 4 elements. Understanding probability in this way helps us grasp how sets and their subsets interact, and it can be applied in various fields such as statistics, computer science, and even everyday decision-making.