# Sets A,B and C are subsets of U.

U= positive integers less than 16
A= prime numbers
B= factors of 36
C= multiples of 4

(A intersect B)' union C {?}

1,4,5,6,7,8,9,10,11,12,13,14,15 union 4,8,12
{4,8,12}

## To be in A intersect B, a number must be both prime and a factor of 36

factors of 36 less than 16:
1 3 4 6 9 12
of those only 1 and 3 are prime
unite those with the multiples of 4
which are 4 8 12
and you get
1 3 4 8 12

## To find the solution to the expression (A intersect B)' union C, we need to follow the steps below:

Step 1: Find the intersection of sets A and B.
A = prime numbers less than 16
B = factors of 36

The prime numbers less than 16 are {2, 3, 5, 7, 11, 13}.
The factors of 36 are {1, 2, 3, 4, 6, 9, 12, 18, 36}.

The intersection of A and B is {2, 3}.

Step 2: Find the complement of the intersection (A intersect B)'.
The complement of {2, 3} consists of all the elements in U (positive integers less than 16) that are not in the intersection.

The positive integers less than 16 are {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}.

So, (A intersect B)' is {1, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}.

Step 3: Find the union of the complement (A intersect B)' and set C.
C = multiples of 4

The multiples of 4 less than 16 are {4, 8, 12}.

The union of (A intersect B)' and C is the combined set of all elements from both sets.

So, the final result is {1, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15} union {4, 8, 12}.

Therefore, the answer is {1, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 4, 8, 12}.

## To find the solution to the expression (A intersect B)' union C, we need to first determine the elements that belong to each set.

Set U is defined as positive integers less than 16, so U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}.

Set A is defined as prime numbers. The prime numbers less than 16 are {2, 3, 5, 7, 11, 13}.

Set B is defined as factors of 36. To find the factors of 36, we can find all the numbers that divide 36 evenly. The factors of 36 are {1, 2, 3, 4, 6, 9, 12, 18, 36}.

Set C is defined as multiples of 4. To find the multiples of 4 less than 16, we can multiply 4 by each positive integer less than 16. The multiples of 4 are {4, 8, 12}.

Now, let's solve the expression (A intersect B)' union C step by step:

Step 1: Find the intersection of sets A and B. The elements that belong to both sets A and B are {2, 3}. Therefore, A intersect B = {2, 3}.

Step 2: Take the complement of the intersection. The complement of {2, 3} in set U (positive integers less than 16) is {1, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}. Therefore, (A intersect B)' = {1, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}.

Step 3: Find the union of the complement (A intersect B)' and set C. The union of {1, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15} and {4, 8, 12} is {1, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 4, 8, 12}.

Therefore, the solution to the expression (A intersect B)' union C is {1, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 4, 8, 12}.