Sets, Venn Diagrams, and Overlap - GRE Quantitative Reasoning
Card 1 of 23
Find $|A\setminus B|$ if $|A|=25$ and $|A\cap B|=9$.
Find $|A\setminus B|$ if $|A|=25$ and $|A\cap B|=9$.
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$16$. Computes set difference as elements only in A: $25 - 9 = 16$.
$16$. Computes set difference as elements only in A: $25 - 9 = 16$.
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What does it mean to say two sets $A$ and $B$ are disjoint?
What does it mean to say two sets $A$ and $B$ are disjoint?
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$A\cap B=\varnothing$. Indicates that A and B have no elements in common, so their intersection is the empty set.
$A\cap B=\varnothing$. Indicates that A and B have no elements in common, so their intersection is the empty set.
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What does it mean to say $A\subseteq B$?
What does it mean to say $A\subseteq B$?
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Every element of $A$ is in $B$. Indicates that A is a subset of B, meaning all elements of A are also elements of B.
Every element of $A$ is in $B$. Indicates that A is a subset of B, meaning all elements of A are also elements of B.
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What is the definition of the set difference $A\setminus B$?
What is the definition of the set difference $A\setminus B$?
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$A\setminus B={x: x\in A\text{ and }x\notin B}$. Defines the set of elements in A but not in B.
$A\setminus B={x: x\in A\text{ and }x\notin B}$. Defines the set of elements in A but not in B.
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What is the definition of the complement $A^c$ relative to universal set $U$?
What is the definition of the complement $A^c$ relative to universal set $U$?
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$A^c=U\setminus A={x\in U: x\notin A}$. Defines the set of elements in the universal set U that are not in A.
$A^c=U\setminus A={x\in U: x\notin A}$. Defines the set of elements in the universal set U that are not in A.
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What is the definition of the intersection $A \cap B$ in set notation?
What is the definition of the intersection $A \cap B$ in set notation?
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$A \cap B={x: x\in A\text{ and }x\in B}$. Defines the set containing all elements that belong to both A and B.
$A \cap B={x: x\in A\text{ and }x\in B}$. Defines the set containing all elements that belong to both A and B.
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What is the definition of the union $A \cup B$ in set notation?
What is the definition of the union $A \cup B$ in set notation?
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$A \cup B={x: x\in A\text{ or }x\in B}$. Defines the set containing all elements that belong to A or B or both.
$A \cup B={x: x\in A\text{ or }x\in B}$. Defines the set containing all elements that belong to A or B or both.
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Find $|A\cap B^c|$ if $|A|=16$ and $|A\cap B|=9$.
Find $|A\cap B^c|$ if $|A|=16$ and $|A\cap B|=9$.
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$7$. Calculates elements in A but not B: $16 - 9 = 7$.
$7$. Calculates elements in A but not B: $16 - 9 = 7$.
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State the formula for the size of a union: $|A\cup B|$ in terms of $|A|,|B|,|A\cap B|$.
State the formula for the size of a union: $|A\cup B|$ in terms of $|A|,|B|,|A\cap B|$.
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$|A\cup B|=|A|+|B|-|A\cap B|$. Applies the inclusion-exclusion principle to count elements in the union without double-counting the intersection.
$|A\cup B|=|A|+|B|-|A\cap B|$. Applies the inclusion-exclusion principle to count elements in the union without double-counting the intersection.
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State De Morgan's law for the complement of a union: $(A\cup B)^c$.
State De Morgan's law for the complement of a union: $(A\cup B)^c$.
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$(A\cup B)^c=A^c\cap B^c$. States De Morgan's law, equating the complement of a union to the intersection of complements.
$(A\cup B)^c=A^c\cap B^c$. States De Morgan's law, equating the complement of a union to the intersection of complements.
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State De Morgan's law for the complement of an intersection: $(A\cap B)^c$.
State De Morgan's law for the complement of an intersection: $(A\cap B)^c$.
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$(A\cap B)^c=A^c\cup B^c$. States De Morgan's law, equating the complement of an intersection to the union of complements.
$(A\cap B)^c=A^c\cup B^c$. States De Morgan's law, equating the complement of an intersection to the union of complements.
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What is the relationship between $A\setminus B$ and $A\cap B^c$?
What is the relationship between $A\setminus B$ and $A\cap B^c$?
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$A\setminus B=A\cap B^c$. Expresses set difference as the intersection of A with the complement of B.
$A\setminus B=A\cap B^c$. Expresses set difference as the intersection of A with the complement of B.
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Identify the set represented by "in exactly one of $A$ or $B$" using set notation.
Identify the set represented by "in exactly one of $A$ or $B$" using set notation.
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$(A\setminus B)\cup(B\setminus A)$. Represents elements that are in exactly one of the sets A or B, forming the symmetric difference.
$(A\setminus B)\cup(B\setminus A)$. Represents elements that are in exactly one of the sets A or B, forming the symmetric difference.
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What is the definition of the symmetric difference $A\triangle B$?
What is the definition of the symmetric difference $A\triangle B$?
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$A\triangle B=(A\setminus B)\cup(B\setminus A)$. Defines the symmetric difference as the union of elements in A but not B and elements in B but not A.
$A\triangle B=(A\setminus B)\cup(B\setminus A)$. Defines the symmetric difference as the union of elements in A but not B and elements in B but not A.
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Find $|A\cup B|$ if $|A|=18$, $|B|=15$, and $|A\cap B|=7$.
Find $|A\cup B|$ if $|A|=18$, $|B|=15$, and $|A\cap B|=7$.
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$26$. Calculates the union size using inclusion-exclusion: $18 + 15 - 7 = 26$.
$26$. Calculates the union size using inclusion-exclusion: $18 + 15 - 7 = 26$.
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Find $|A\cap B|$ if $|A|=20$, $|B|=17$, and $|A\cup B|=30$.
Find $|A\cap B|$ if $|A|=20$, $|B|=17$, and $|A\cup B|=30$.
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$7$. Solves for intersection using inclusion-exclusion rearranged: $20 + 17 - 30 = 7$.
$7$. Solves for intersection using inclusion-exclusion rearranged: $20 + 17 - 30 = 7$.
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Find $|B\setminus A|$ if $|B|=19$ and $|A\cap B|=6$.
Find $|B\setminus A|$ if $|B|=19$ and $|A\cap B|=6$.
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$13$. Computes set difference as elements only in B: $19 - 6 = 13$.
$13$. Computes set difference as elements only in B: $19 - 6 = 13$.
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Find $|A^c|$ if $|U|=60$ and $|A|=22$.
Find $|A^c|$ if $|U|=60$ and $|A|=22$.
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$38$. Calculates the complement as elements not in A: $60 - 22 = 38$.
$38$. Calculates the complement as elements not in A: $60 - 22 = 38$.
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Find $|A\cup B|$ if $|U|=50$ and $|(A\cup B)^c|=12$.
Find $|A\cup B|$ if $|U|=50$ and $|(A\cup B)^c|=12$.
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$38$. Determines union size by subtracting the complement from the universal set: $50 - 12 = 38$.
$38$. Determines union size by subtracting the complement from the universal set: $50 - 12 = 38$.
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Find $|A\triangle B|$ if $|A|=14$, $|B|=11$, and $|A\cap B|=5$.
Find $|A\triangle B|$ if $|A|=14$, $|B|=11$, and $|A\cap B|=5$.
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$15$. Computes symmetric difference as sum of exclusive parts: $(14-5) + (11-5) = 15$.
$15$. Computes symmetric difference as sum of exclusive parts: $(14-5) + (11-5) = 15$.
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State the 3-set inclusion–exclusion formula for $|A\cup B\cup C|$.
State the 3-set inclusion–exclusion formula for $|A\cup B\cup C|$.
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$|A\cup B\cup C|=|A|+|B|+|C|-|A\cap B|-|A\cap C|-|B\cap C|+|A\cap B\cap C|$. Applies the inclusion-exclusion principle for three sets to count the total unique elements in the union.
$|A\cup B\cup C|=|A|+|B|+|C|-|A\cap B|-|A\cap C|-|B\cap C|+|A\cap B\cap C|$. Applies the inclusion-exclusion principle for three sets to count the total unique elements in the union.
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Find $|A\cap B\cap C|$ if $|A\cap B|=9$, $|A\cap B\setminus C|=6$.
Find $|A\cap B\cap C|$ if $|A\cap B|=9$, $|A\cap B\setminus C|=6$.
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$3$. Determines the triple intersection by subtracting the part excluding C: $9-6=3$.
$3$. Determines the triple intersection by subtracting the part excluding C: $9-6=3$.
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Find $|U\setminus(A\cup B)|$ if $|U|=80$, $|A|=35$, $|B|=30$, and $|A\cap B|=10$.
Find $|U\setminus(A\cup B)|$ if $|U|=80$, $|A|=35$, $|B|=30$, and $|A\cap B|=10$.
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$25$. Calculates elements outside the union: $80 - (35+30-10)=25$.
$25$. Calculates elements outside the union: $80 - (35+30-10)=25$.
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