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# Set Operations

Familiarize yourself with the foundational notation, tools and concepts for the operations that are applied to sets.

# Set Operations: Level 2 Challenges

If $$A$$ and $$B$$ are subsets of $$U=\{1,2,3,4,5\}$$ such that $A \cup B = U, A\cap B =\emptyset,$ how many possible choices are there for the ordered pair $$(A,B)$$?

Details and assumptions

It is possible that $$A = \emptyset$$ or $$B = \emptyset$$.

Let $$A$$ and $$B$$ be two sets and $$U$$ be a universal set such that $$|U|=700$$, $$|A|=200$$, $$|B|=300$$ and $$|A\cap B|=100$$. Find $$|A^{C}\cap B^{C}|$$.

Which of the following sequence of set operations counts the number of elements that are in exactly one of $$A$$, $$B$$ or $$C$$?"

Options:

1. $$|A\cap B|+|B\cap C|+|C\cap A|-3(|A\cap B\cap C|)$$

2. $$|A|+|B|+|C|-2(|A\cap B|+|B\cap C|+|C\cap A|)+3|A\cap B\cap C|$$

3. $$|A\cap B|+|B\cap C|+|C\cap A|-2|A\cap B\cap C|$$

4. $$|A|+|B|+|C|-|A\cap B|-|B\cap C|-|C\cap A|+|A\cap B\cap C|$$

5. $$|(A\cap B\cap C)\cup(A\cap B^{C}\cap C^{C})\cup(A^{C}\cap B\cap C^{C})\cup(A^{C}\cap B^{C}\cap C)|$$

6. |$$A\cup B\cup C$$ |

If X and Y are two sets, X $$∩$$ $$(Y∪X)^c$$ is equal to

Details and assumptions:
$$\rightarrow$$ $$∩$$ represents intersection.
$$\rightarrow$$ $$∪$$ represents union.
$$\rightarrow$$ $$X^c$$ represents compliment of set X.

Person "A" says the truth 60% of the time, and person "B" does so 90% of the time. In what percentage of cases are they likely to contradict each other in stating the same fact?

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