We are given two sets $X$ and $Y$ with the following information:

• $X_1$, a subset of $X$, has 12 elements: $|X_1| = 12$.
• $Y_1$, a subset of $Y$, has 7 elements: $|Y_1| = 7$.
• The intersection of $X_1$ and $Y_1$ contains 4 elements: $|X_1 \cap Y_1| = 4$.

The task is to determine the number of elements in $X' \cap Y$, where $X'$ represents the complement of $X_1$, i.e., $X' = X \setminus X_1$.

Step-by-Step Solution:

We are interested in the number of elements in $X' \cap Y$, where:

$X' \cap Y = (X \setminus X_1) \cap Y$

However, we only have information about the subset $X_1$, not the entire set $X$ or $Y$. But, focusing on subsets, we can consider how to work with complements and intersections.

1. Since $X'$ is the complement of $X_1$, any element in $X' \cap Y$ will not belong to $X_1$.
2. $|X_1 \cap Y_1| = 4$ elements are shared between $X_1$ and $Y_1$. These are excluded from $X'$.
3. The remaining elements in $Y_1 \setminus X_1$ (which belong to $Y_1$ but not to $X_1$) form part of $X' \cap Y$.

Thus, the number of elements in $X' \cap Y$ is:

$|X' \cap Y| = |Y_1| - |X_1 \cap Y_1| = 7 - 4 = 3$

Final Answer:

The number of elements in $X' \cap Y$ is 3.

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Do you have any questions or need further details?

Here are 5 related questions for further exploration:

1. What is the cardinality of $X \cup Y_1$?
2. How would the result change if $X_1 \cap Y_1$ had more or fewer elements?
3. Can you determine $|X \cap Y|$ if more information about $X$ or $Y$ is given?
4. How do set complements behave when working with larger sets?
5. What is the relationship between $X_1 \setminus Y_1$ and $Y_1 \setminus X_1$?

Tip: Understanding how set intersections and complements behave is fundamental to solving problems involving subsets and their relationships.