Sets - GMAT Quantitative

Card 0 of 160

Question

$A = \left \{ 2, 4, 6, 8, 10, 12, ...\right \}$

$B = \left \{ 3, 6, 9, 12, 15, 18...\right \}$

True or false: $n \in A \cup B$

Statement 1: is a perfect square.

Statement 2: is a multiple of 99.

Answer

includes all multiples of 2; includes all multiples of 3. $A \cup B$ comprises all multiples of either 2 or 3.

Knowing is a perfect square is neither necessary nor helpful, as, for example, $9 \in B \subseteq A \cup B$ , but $25 \notin A \cup B$ (as 25 is neither a multiple of 2 nor a multiple of 3).

If you know that is a multiple of 99, then it must also be a multiple of any number that divides 99 evenly, one such number is 3. This means $n \in B \subseteq A \cup B$

Compare your answer with the correct one above

Question

$A = \left \{ 2, 4, 6, 8, 10, 12, ...\right \}$

$B = \left \{ 3, 6, 9, 12, 15, 18...\right \}$

True or false: $n \in A \cup B$

Statement 1: is a perfect square.

Statement 2: is a multiple of 99.

Answer

includes all multiples of 2; includes all multiples of 3. $A \cup B$ comprises all multiples of either 2 or 3.

Knowing is a perfect square is neither necessary nor helpful, as, for example, $9 \in B \subseteq A \cup B$ , but $25 \notin A \cup B$ (as 25 is neither a multiple of 2 nor a multiple of 3).

If you know that is a multiple of 99, then it must also be a multiple of any number that divides 99 evenly, one such number is 3. This means $n \in B \subseteq A \cup B$

Compare your answer with the correct one above

Question

$A = \left \{ 2, 4, 6, 8, 10, 12, ...\right \}$

$B = \left \{ 3, 6, 9, 12, 15, 18...\right \}$

True or false: $n \in A \cup B$

Statement 1: is a perfect square.

Statement 2: is a multiple of 99.

Answer

includes all multiples of 2; includes all multiples of 3. $A \cup B$ comprises all multiples of either 2 or 3.

Knowing is a perfect square is neither necessary nor helpful, as, for example, $9 \in B \subseteq A \cup B$ , but $25 \notin A \cup B$ (as 25 is neither a multiple of 2 nor a multiple of 3).

If you know that is a multiple of 99, then it must also be a multiple of any number that divides 99 evenly, one such number is 3. This means $n \in B \subseteq A \cup B$

Compare your answer with the correct one above

Question

$A = \left \{ 2, 4, 6, 8, 10, 12, ...\right \}$

$B = \left \{ 3, 6, 9, 12, 15, 18...\right \}$

True or false: $n \in A \cup B$

Statement 1: is a perfect square.

Statement 2: is a multiple of 99.

Answer

includes all multiples of 2; includes all multiples of 3. $A \cup B$ comprises all multiples of either 2 or 3.

Knowing is a perfect square is neither necessary nor helpful, as, for example, $9 \in B \subseteq A \cup B$ , but $25 \notin A \cup B$ (as 25 is neither a multiple of 2 nor a multiple of 3).

If you know that is a multiple of 99, then it must also be a multiple of any number that divides 99 evenly, one such number is 3. This means $n \in B \subseteq A \cup B$

Compare your answer with the correct one above

Question

$A = \left \{ 2, 4, 6, 8, 10, 12, ...\right \}$

$B = \left \{ 3, 6, 9, 12, 15, 18...\right \}$

True or false: $n \in A \cup B$

Statement 1: is a perfect square.

Statement 2: is a multiple of 99.

Answer

includes all multiples of 2; includes all multiples of 3. $A \cup B$ comprises all multiples of either 2 or 3.

Knowing is a perfect square is neither necessary nor helpful, as, for example, $9 \in B \subseteq A \cup B$ , but $25 \notin A \cup B$ (as 25 is neither a multiple of 2 nor a multiple of 3).

If you know that is a multiple of 99, then it must also be a multiple of any number that divides 99 evenly, one such number is 3. This means $n \in B \subseteq A \cup B$

Compare your answer with the correct one above

Question

$A = \left \{ 2, 4, 6, 8, 10, 12, ...\right \}$

$B = \left \{ 3, 6, 9, 12, 15, 18...\right \}$

True or false: $n \in A \cup B$

Statement 1: is a perfect square.

Statement 2: is a multiple of 99.

Answer

includes all multiples of 2; includes all multiples of 3. $A \cup B$ comprises all multiples of either 2 or 3.

Knowing is a perfect square is neither necessary nor helpful, as, for example, $9 \in B \subseteq A \cup B$ , but $25 \notin A \cup B$ (as 25 is neither a multiple of 2 nor a multiple of 3).

If you know that is a multiple of 99, then it must also be a multiple of any number that divides 99 evenly, one such number is 3. This means $n \in B \subseteq A \cup B$

Compare your answer with the correct one above

Question

$A = \left \{ 2, 4, 6, 8, 10, 12, ...\right \}$

$B = \left \{ 3, 6, 9, 12, 15, 18...\right \}$

True or false: $n \in A \cup B$

Statement 1: is a perfect square.

Statement 2: is a multiple of 99.

Answer

includes all multiples of 2; includes all multiples of 3. $A \cup B$ comprises all multiples of either 2 or 3.

Knowing is a perfect square is neither necessary nor helpful, as, for example, $9 \in B \subseteq A \cup B$ , but $25 \notin A \cup B$ (as 25 is neither a multiple of 2 nor a multiple of 3).

If you know that is a multiple of 99, then it must also be a multiple of any number that divides 99 evenly, one such number is 3. This means $n \in B \subseteq A \cup B$

Compare your answer with the correct one above

Question

Define two sets as follows:

$A = \left \{2, 4, 6, 8, 10, a, b \right \}$

$B = \left \{1, 3, 5, 7, 9, c, d \right \}$

where and are distinct positive odd integers and and are distinct positive even integers.

How many elements are contained in the set $A \cap B$ ?

$a, b \in \left \{ 1, 3, 5 ,7, 9\right \}$

$c,d \in \left \{ 2, 4, 6, 8, 10\right \}$

Answer

Suppose we know $a, b \in \left \{ 1, 3, 5 ,7, 9\right \}$ , but we do not assume the second statement.

If and , then $A \cap B = \left \{}a, b, 2, 4 \right \}$ , a four-element set. If If and , then $A \cap B = \left \{}a, b, 2 \right \}$ , a three-element set. Therefore, we cannot make a conclusion about the size of $A \cap B$ . A similar argument can be used to show that assuming only the second statement also does not allow a conclusion.

If we know both statements, however, $A \cap B = \left \{}a, b, c, d \right \}$ , and we can prove that $A \cap B$ has four elements.

The answer is that both statements together are sufficient to answer this question, but neither statement alone is sufficient.

Compare your answer with the correct one above

Question

$A = \left \{ 2, 4, 6, 8, 10, 12, ...\right \}$

$B = \left \{ 3, 6, 9, 12, 15, 18...\right \}$

True or false: $n \in A \cup B$

Statement 1: is a perfect square.

Statement 2: is a multiple of 99.

Answer

includes all multiples of 2; includes all multiples of 3. $A \cup B$ comprises all multiples of either 2 or 3.

Knowing is a perfect square is neither necessary nor helpful, as, for example, $9 \in B \subseteq A \cup B$ , but $25 \notin A \cup B$ (as 25 is neither a multiple of 2 nor a multiple of 3).

If you know that is a multiple of 99, then it must also be a multiple of any number that divides 99 evenly, one such number is 3. This means $n \in B \subseteq A \cup B$

Compare your answer with the correct one above

Question

Define two sets as follows:

$A = \left \{2, 4, 6, 8, 10, a, b \right \}$

$B = \left \{1, 3, 5, 7, 9, c, d \right \}$

where and are distinct positive odd integers and and are distinct positive even integers.

How many elements are contained in the set $A \cap B$ ?

$a, b \in \left \{ 1, 3, 5 ,7, 9\right \}$

$c,d \in \left \{ 2, 4, 6, 8, 10\right \}$

Answer

Suppose we know $a, b \in \left \{ 1, 3, 5 ,7, 9\right \}$ , but we do not assume the second statement.

If and , then $A \cap B = \left \{}a, b, 2, 4 \right \}$ , a four-element set. If If and , then $A \cap B = \left \{}a, b, 2 \right \}$ , a three-element set. Therefore, we cannot make a conclusion about the size of $A \cap B$ . A similar argument can be used to show that assuming only the second statement also does not allow a conclusion.

If we know both statements, however, $A \cap B = \left \{}a, b, c, d \right \}$ , and we can prove that $A \cap B$ has four elements.

The answer is that both statements together are sufficient to answer this question, but neither statement alone is sufficient.

Compare your answer with the correct one above

Question

How many elements are in the set $A \cup B$ ?

Statement 1: has three more elements than .

Statement 2: includes exactly four elements not in .

Answer

Assume both statements are true.

Consider these two cases:

Case1: $A = \left \{ 1,2,3,4,5\right \}$ and $B = \left \{ 5,6\right \}$

Case 2: $A = \left \{ 1,2,3,4,5,6\right \}$ and $B = \left \{ 5,6,7\right \}$

In both situations, has three more elements than and includes exactly four elements not in (1, 2, 3 and 4). However, the number of elements in the union differ in each case - in the first case, $A \cup B= \left \{ 1,2,3,4,5,6\right \}$ , and in the second case, $A \cup B= \left \{ 1,2,3,4,5,6, 7\right \}$ .

The two statements together do not yield an answer to the question.

Compare your answer with the correct one above

Question

How many elements are in the set $A \cup B$ ?

Statement 1: has three more elements than .

Statement 2: includes exactly four elements not in .

Answer

Assume both statements are true.

Consider these two cases:

Case1: $A = \left \{ 1,2,3,4,5\right \}$ and $B = \left \{ 5,6\right \}$

Case 2: $A = \left \{ 1,2,3,4,5,6\right \}$ and $B = \left \{ 5,6,7\right \}$

In both situations, has three more elements than and includes exactly four elements not in (1, 2, 3 and 4). However, the number of elements in the union differ in each case - in the first case, $A \cup B= \left \{ 1,2,3,4,5,6\right \}$ , and in the second case, $A \cup B= \left \{ 1,2,3,4,5,6, 7\right \}$ .

The two statements together do not yield an answer to the question.

Compare your answer with the correct one above

Question

How many elements are in the set $A \cup B$ ?

Statement 1: has three more elements than .

Statement 2: includes exactly four elements not in .

Answer

Assume both statements are true.

Consider these two cases:

Case1: $A = \left \{ 1,2,3,4,5\right \}$ and $B = \left \{ 5,6\right \}$

Case 2: $A = \left \{ 1,2,3,4,5,6\right \}$ and $B = \left \{ 5,6,7\right \}$

In both situations, has three more elements than and includes exactly four elements not in (1, 2, 3 and 4). However, the number of elements in the union differ in each case - in the first case, $A \cup B= \left \{ 1,2,3,4,5,6\right \}$ , and in the second case, $A \cup B= \left \{ 1,2,3,4,5,6, 7\right \}$ .

The two statements together do not yield an answer to the question.

Compare your answer with the correct one above

Question

How many elements are in the set $A \cup B$ ?

Statement 1: has three more elements than .

Statement 2: includes exactly four elements not in .

Answer

Assume both statements are true.

Consider these two cases:

Case1: $A = \left \{ 1,2,3,4,5\right \}$ and $B = \left \{ 5,6\right \}$

Case 2: $A = \left \{ 1,2,3,4,5,6\right \}$ and $B = \left \{ 5,6,7\right \}$

In both situations, has three more elements than and includes exactly four elements not in (1, 2, 3 and 4). However, the number of elements in the union differ in each case - in the first case, $A \cup B= \left \{ 1,2,3,4,5,6\right \}$ , and in the second case, $A \cup B= \left \{ 1,2,3,4,5,6, 7\right \}$ .

The two statements together do not yield an answer to the question.

Compare your answer with the correct one above

Question

How many elements are in the set $A \cup B$ ?

Statement 1: has three more elements than .

Statement 2: includes exactly four elements not in .

Answer

Assume both statements are true.

Consider these two cases:

Case1: $A = \left \{ 1,2,3,4,5\right \}$ and $B = \left \{ 5,6\right \}$

Case 2: $A = \left \{ 1,2,3,4,5,6\right \}$ and $B = \left \{ 5,6,7\right \}$

In both situations, has three more elements than and includes exactly four elements not in (1, 2, 3 and 4). However, the number of elements in the union differ in each case - in the first case, $A \cup B= \left \{ 1,2,3,4,5,6\right \}$ , and in the second case, $A \cup B= \left \{ 1,2,3,4,5,6, 7\right \}$ .

The two statements together do not yield an answer to the question.

Compare your answer with the correct one above

Question

How many elements are in the set $A \cup B$ ?

Statement 1: has three more elements than .

Statement 2: includes exactly four elements not in .

Answer

Assume both statements are true.

Consider these two cases:

Case1: $A = \left \{ 1,2,3,4,5\right \}$ and $B = \left \{ 5,6\right \}$

Case 2: $A = \left \{ 1,2,3,4,5,6\right \}$ and $B = \left \{ 5,6,7\right \}$

In both situations, has three more elements than and includes exactly four elements not in (1, 2, 3 and 4). However, the number of elements in the union differ in each case - in the first case, $A \cup B= \left \{ 1,2,3,4,5,6\right \}$ , and in the second case, $A \cup B= \left \{ 1,2,3,4,5,6, 7\right \}$ .

The two statements together do not yield an answer to the question.

Compare your answer with the correct one above

Question

How many elements are in the set $A \cup B$ ?

Statement 1: has three more elements than .

Statement 2: includes exactly four elements not in .

Answer

Assume both statements are true.

Consider these two cases:

Case1: $A = \left \{ 1,2,3,4,5\right \}$ and $B = \left \{ 5,6\right \}$

Case 2: $A = \left \{ 1,2,3,4,5,6\right \}$ and $B = \left \{ 5,6,7\right \}$

In both situations, has three more elements than and includes exactly four elements not in (1, 2, 3 and 4). However, the number of elements in the union differ in each case - in the first case, $A \cup B= \left \{ 1,2,3,4,5,6\right \}$ , and in the second case, $A \cup B= \left \{ 1,2,3,4,5,6, 7\right \}$ .

The two statements together do not yield an answer to the question.

Compare your answer with the correct one above

Question

Define two sets as follows:

$A = \left \{2, 4, 6, 8, 10, a, b \right \}$

$B = \left \{1, 3, 5, 7, 9, c, d \right \}$

where and are distinct positive odd integers and and are distinct positive even integers.

How many elements are contained in the set $A \cap B$ ?

$a, b \in \left \{ 1, 3, 5 ,7, 9\right \}$

$c,d \in \left \{ 2, 4, 6, 8, 10\right \}$

Answer

Suppose we know $a, b \in \left \{ 1, 3, 5 ,7, 9\right \}$ , but we do not assume the second statement.

If and , then $A \cap B = \left \{}a, b, 2, 4 \right \}$ , a four-element set. If If and , then $A \cap B = \left \{}a, b, 2 \right \}$ , a three-element set. Therefore, we cannot make a conclusion about the size of $A \cap B$ . A similar argument can be used to show that assuming only the second statement also does not allow a conclusion.

If we know both statements, however, $A \cap B = \left \{}a, b, c, d \right \}$ , and we can prove that $A \cap B$ has four elements.

The answer is that both statements together are sufficient to answer this question, but neither statement alone is sufficient.

Compare your answer with the correct one above

Question

How many elements are in the set $A \cup B$ ?

Statement 1: has three more elements than .

Statement 2: includes exactly four elements not in .

Answer

Assume both statements are true.

Consider these two cases:

Case1: $A = \left \{ 1,2,3,4,5\right \}$ and $B = \left \{ 5,6\right \}$

Case 2: $A = \left \{ 1,2,3,4,5,6\right \}$ and $B = \left \{ 5,6,7\right \}$

In both situations, has three more elements than and includes exactly four elements not in (1, 2, 3 and 4). However, the number of elements in the union differ in each case - in the first case, $A \cup B= \left \{ 1,2,3,4,5,6\right \}$ , and in the second case, $A \cup B= \left \{ 1,2,3,4,5,6, 7\right \}$ .

The two statements together do not yield an answer to the question.

Compare your answer with the correct one above

Question

Define two sets as follows:

$A = \left \{2, 4, 6, 8, 10, a, b \right \}$

$B = \left \{1, 3, 5, 7, 9, c, d \right \}$

where and are distinct positive odd integers and and are distinct positive even integers.

How many elements are contained in the set $A \cap B$ ?

$a, b \in \left \{ 1, 3, 5 ,7, 9\right \}$

$c,d \in \left \{ 2, 4, 6, 8, 10\right \}$

Answer

Suppose we know $a, b \in \left \{ 1, 3, 5 ,7, 9\right \}$ , but we do not assume the second statement.

If and , then $A \cap B = \left \{}a, b, 2, 4 \right \}$ , a four-element set. If If and , then $A \cap B = \left \{}a, b, 2 \right \}$ , a three-element set. Therefore, we cannot make a conclusion about the size of $A \cap B$ . A similar argument can be used to show that assuming only the second statement also does not allow a conclusion.

If we know both statements, however, $A \cap B = \left \{}a, b, c, d \right \}$ , and we can prove that $A \cap B$ has four elements.

The answer is that both statements together are sufficient to answer this question, but neither statement alone is sufficient.

Compare your answer with the correct one above

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