The problem states:

Set $A$ has $a$ elements and set $B$ has $b$ elements. The sum of the number of proper subsets of $A$ and $B$ is 142. We need to find the value of $a - b$.

Solution:

Step 1: Formula for the number of proper subsets

The number of subsets of a set with $n$ elements is $2^n$. The number of proper subsets is: $2^n - 1$ (since we exclude the set itself).

Step 2: Use the given condition

The number of proper subsets of $A$ is $2^a - 1$, and the number of proper subsets of $B$ is $2^b - 1$.

The sum of the proper subsets is given as: $(2^a - 1) + (2^b - 1) = 142$ Simplify: $2^a + 2^b - 2 = 142$ $2^a + 2^b = 144$

Step 3: Solve for $a$ and $b$

We now find integers $a$ and $b$ such that: $2^a + 2^b = 144$

• Test possible values:
  • $a = 7$: $2^7 = 128$, so $2^b = 144 - 128 = 16$ → $b = 4$.

Thus: $a = 7, \, b = 4$

Step 4: Calculate $a - b$

$a - b = 7 - 4 = 3$

Final Answer:

$a - b = 3$

Would you like further explanation or examples? Here are related questions to explore:

1. How do we derive the formula for the number of subsets of a set?
2. Can you solve a similar problem with different numbers of subsets?
3. What is the difference between subsets and proper subsets in set theory?
4. How does this problem relate to binary representations of sets?
5. How do exponential growth rates affect problems involving sets?

Tip: Always verify your solution by substituting values back into the original equation!