To determine which statement is true, let us analyze each option using set theory principles.

Definitions:

1. $|A|$: The number of elements in set $A$.
2. $|B|$: The number of elements in set $B$.
3. $|A \cap B|$: The number of elements in the intersection of sets $A$ and $B$ (elements common to both).
4. $|A \cup B|$: The number of elements in the union of sets $A$ and $B$ (elements in either $A$, $B$, or both).

Analyzing Each Option:

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a. $|A \cup B| = |A| + |B| + |A \cap B|$

This is false. The correct formula for $|A \cup B|$ is: $|A \cup B| = |A| + |B| - |A \cap B|$ Adding $|A \cap B|$ here would incorrectly double-count the elements in the intersection.

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b. $|A \cap B| = |A| + |B| - |A \cup B|$

This is true. The formula for the size of the intersection of two sets is: $|A \cap B| = |A| + |B| - |A \cup B|$ This ensures that elements in the union are counted correctly without duplication.

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c. $|A \cup B| = |A| + |B|$

This is false. Again, the formula for $|A \cup B|$ is: $|A \cup B| = |A| + |B| - |A \cap B|$ This formula accounts for the overlap between sets $A$ and $B$. The given statement does not account for the intersection and is incorrect.

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d. $|A \cap B| = |A| - |B|$

This is false. The size of the intersection of two sets has no simple relationship of subtraction like this. The correct formula for the intersection involves the union as in option (b).

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Correct Answer:

b. $|A \cap B| = |A| + |B| - |A \cup B|$

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Do you want further details or examples on this?

Related Questions:

1. How does the principle of inclusion-exclusion apply to sets?
2. Can you provide an example using real sets $A$ and $B$?
3. What happens when $|A \cap B| = 0$?
4. How do we generalize this formula for more than two sets?
5. Can you explain the Venn diagram interpretation of these formulas?

Tip:

Remember that the union formula is derived from avoiding double-counting elements in the intersection. Always consider overlaps when combining sets!