Set Theory

We are given the following information: We need to find .

1. Step 1: Use the formula for the number of elements in a cross-product The number of elements in the cross-product of two sets is the product of the number of elements in each set:

2. Step 2: Set up the system of equations From the given information, we have:

3. Step 3: Solve the system of equations To find , we first eliminate and . Multiply the first and second equations: Simplifying this: Using the third equation , substitute this into the equation: Solving for : We can now solve for .

The number of relations from a set to a set is given by: We are given that and the number of relations is 128. Therefore, we have: Solving for :
Thus, . Hence, the correct answer is .

Concept:
Set Theory - Principle of Inclusion-Exclusion.
Step 1: Define the sets.
Let be the set of positive integers up to 100 divisible by 3.
Let be the set of positive integers up to 100 divisible by 7.
The target range is .
Step 2: Find the number of integers divisible by 3.
To find the number of elements in set , divide the maximum limit by 3 and take the greatest integer (floor function): $ B A \cap B $

Concept:
Relation and Functions - Reflexive Relation.
Step 1: Recall the definition of a reflexive relation.
A relation on a set is said to be reflexive if every element in is related to itself.
Mathematically, for all .
Step 2: Identify the required pairs for the given set X.
Given the set .
For to be reflexive, it MUST contain the following pairs: $ R = \{(1,1), (1, 2), (1, 3), (5, 6), (6, 7), (7,5)\} (1,1) R R$ reflexive.

Concept:
To evaluate the number of possible subsets for , we must use the definition of set difference:

• consists of elements that are in but NOT in .

Step 1: Identify the given sets and the conditions for set B.
Given the universal set and set : $ B \subseteq X A \setminus B = \{d\} d A B a, b, c A A \setminus B B e X A A e B A \setminus B e B B \{a, b, c\} e P P $