Let 𝐴={1,2,3,4,5,6,7,8,9}𝐴={1,2,3,4,5,6,7,8,9} and 𝐵𝐵 be the set of primes. What is |𝐴∩(ℤ−𝐵)||𝐴∩(𝑍−𝐵)|?

First, let's break down the question:

1. Set A is given as {1,2,3,4,5,6,7,8,9}.
2. Set B is the set of prime numbers.
3. ℤ represents the set of all integers.
4. We are asked to find the cardinality (or size) of the intersection of set A and the difference between ℤ and B. This is represented as |𝐴∩(ℤ−𝐵)|.

Now, let's solve the problem step by step:

Step 1: Identify the prime numbers in set A. These are {2,3,5,7}.

Step 2: Subtract these primes from set A to get the set of non-prime numbers in A. This gives us {1,4,6,8,9}.

Step 3: The question asks for the intersection of set A and (ℤ - B). Since ℤ represents all integers and B is the set of all primes, (ℤ - B) represents all non-prime integers.

Step 4: The intersection of set A and all non-prime integers is simply the set of non-prime numbers in A, which we found in step 2.

So, |𝐴∩(ℤ−𝐵)| is the number of elements in the set {1,4,6,8,9}, which is 5.