Set Theory) Mastery Hub: The Industry Foundation Practice Te

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Q1Domain Verified

Consider two finite sets, A and B, where $|A| = 10$ and $|B| = 15$. If $|A \cup B| = 20$, what is the cardinality of the intersection of A and B, i.e., $|A \cap B|$?

{ "letter": "A", "text": "5" }

{ "letter": "B", "text": "10" }

{ "letter": "C", "text": "15" }

{ "letter": "D", "text": "25" }

Q2Domain Verified

Let U be the universal set, and let A and B be subsets of U. If $A \subseteq B$, which of the following statements is necessarily true?

{ "letter": "A", "text": "$A \\cup B = A$" }

{ "letter": "B", "text": "$A \\cap B = \\emptyset$" }

{ "letter": "C", "text": "$A' \\cup B' = U$" }

{ "letter": "D", "text": "$A \\cap B = B$" }

{ "letter": "A", "text": "$A \\cup B = B$: The union of A and B would be the larger set, B. Thus, $A \\cup B = A$ is incorrect." }

{ "letter": "B", "text": "$A \\cap B = A$: The intersection of A and B would be the smaller set, A. Thus, $A \\cap B = \\emptyset$ is incorrect unless A is the empty set." }

{ "letter": "C", "text": "$A' \\cup B' = U$: By De Morgan's Laws, $A' \\cup B' = (A \\cap B)'$. Since $A \\subseteq B$, we know $A \\cap B = A$. Therefore, $(A \\cap B)' = A'$. This statement is incorrect. Let's re-evaluate. If $A \\subseteq B$, then $A \\cap B = A$. So $(A \\cap B)' = A'$. The statement is $A' \\cup B' = U$. Let's use another De Morgan's Law: $A' \\cap B' = (A \\cup B)'$. If $A \\subseteq B$, then $A \\cup B = B$. So $A' \\cap B' = B'$. This doesn't directly help. Let's consider the contrapositive of the given condition. If $x \\in A$, then $x \\in B$. If $x \\notin B$, then $x \\notin A$. This means $B' \\subseteq A'$." }

{ "letter": "A", "text": "$A \\cup B = \\{1, 2, 3\\} = B$. So $A \\cup B = A$ is false." }

{ "letter": "B", "text": "$A \\cap B = \\{1, 2\\} = A$. So $A \\cap B = \\emptyset$ is false." }

{ "letter": "C", "text": "$A' \\cup B' = \\{3, 4, 5\\} \\cup \\{4, 5\\} = \\{3, 4, 5\\} = A'$. This is not U." }

{ "letter": "D", "text": "$A \\cap B = \\{1, 2\\} = A$. So $A \\cap B = B$ is false." }

Q3Domain Verified

$A \cup B = B$ (This is necessarily true)

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Set Theory) Mastery Hub: The Industry Foundation Practice Te

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Elite Practice Intelligence

Consider two finite sets, A and B, where $|A| = 10$ and $|B| = 15$. If $|A \cup B| = 20$, what is the cardinality of the intersection of A and B, i.e., $|A \cap B|$?

Let U be the universal set, and let A and B be subsets of U. If $A \subseteq B$, which of the following statements is necessarily true?

$A \cup B = B$ (This is necessarily true)

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Set Theory) Mastery Hub: The Industry Foundation Practice Te

Average Pass Rate

Elite Practice Intelligence

Consider two finite sets, A and B, where $|A| = 10$ and $|B| = 15$. If $|A \cup B| = 20$, what is the cardinality of the intersection of A and B, i.e., $|A \cap B|$?

Let U be the universal set, and let A and B be subsets of U. If $A \subseteq B$, which of the following statements is *necessarily* true?

$A \cup B = B$ (This is necessarily true)

Master the Entire Curriculum

Strategic Focus

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1Consider two finite sets, A and B, where $|A| = 10$ and $|B| = 15$. If $|A \cup B| = 20$, what is the cardinality of the intersection of A and B, i.e., $|A \cap B|$?

2Let U be the universal set, and let A and B be subsets of U. If $A \subseteq B$, which of the following statements is *necessarily* true?

3$A \cup B = B$ (This is necessarily true)

Other Recommended Specializations

Mastery: SNAP

Reading Comprehension Strategies Mastery Hub: The Industry Foundation

Vocabulary Building & Usage Mastery Hub: The Industry Foundation

Grammar & Sentence Correction Mastery Hub: The Industry Foundation

Para Jumbles & Odd One Out Mastery Hub: The Industry Foundation

Verbal Analogy & Critical Reasoning Mastery Hub: The Industry Foundation

Let U be the universal set, and let A and B be subsets of U. If $A \subseteq B$, which of the following statements is necessarily true?