GMAT Math : Understanding sets

Study concepts, example questions & explanations for GMAT Math

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Example Questions

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Example Question #1 : Problem Solving Questions

A fair coin is flipped successively until heads are observed on 2 successive flips. Let x denote the number of coin flips required. What is the sample space of x?

Possible Answers:

{: x = 2, 3, 4 . . .}

{: x = 2, 3, 4, 5, 6}

{: x = 0, 1, 2, 3, 4 . . .}

{x : x is a real number}

not enough information

Correct answer:

{: x = 2, 3, 4 . . .}

Explanation:

We need to flip a coin until we get two heads in a row. The smallest number of possible flips is 2, which would occur if our first two flips are both heads. This eliminates three of our answer choices, because we know the sample space must start at 2. 

This leaves us with {: x = 2, 3, 4 . . .} and {x : x = 2, 3, 4, 5, 6}. Let's think about {: x = 2, 3, 4, 5, 6}. What if I flip a coin 6 times and get 6 tails? Then I have to keep flipping beyond 6 flips until I get two heads in a row; therefore the answer must be {x : x = 2, 3, 4 . . .}, because we don't have an upper limit on the number of flips it will take to produce two successive heads.

Example Question #1 : Sets

Which of these Venn diagrams represents the set  ?

Possible Answers:

Venn_3

Venn_5

 

Venn_4

 

Venn_1

 

Venn_2

 

Correct answer:

Venn_1

 

Explanation:

 is the set of elements that fall either in  or the complement of , or both - that is, either in  , outside of , or both. This union is intersected with the complement of , meaning that only the elements of the union that also fall outside of  are considered.

"Color" in all of  and everything outside of  - but then, uncolor everything inside . That makes the correct choice:

Venn_1

Example Question #2 : Sets

Venn

The above represents a Venn diagram. The universal set  is the set of all positive integers.

Let  be the set of all multiples of 3; let  be the set of all multiples of 5; let  be the set of all multiples of 7. Which of the five marked regions would include the number 525?

Possible Answers:

Correct answer:

Explanation:

525 is a multiple of all three of the integers 3, 5, and 7:

Therefore, 525 is an element of each of sets , and, subsequently, falls into region , which represents .

Example Question #4 : Problem Solving Questions

Mark will hire 5 of the 8 job applicants he interviews. In how many different ways can he do this?

Possible Answers:

Correct answer:

Explanation:

Since order doesn't matter here, set this up as a combination:

Example Question #5 : Problem Solving Questions

Venn

Refer to the Venn diagram. Let universal set  be the set of all natural numbers, 

Let  be the set of all multiples of ; let  be the set of all perfect squares; let  be the set of all perfect cubes. Which region of the Venn diagram contains the number ?

Possible Answers:

Correct answer:

Explanation:

, making 1,728 a multiple of 3, and thus, an element of .

1,728 is not a perfect square; . Thus, 1,728 is not an element of .

1,728 is a perfect cube: . Thus, 1,728 is an element of .

, which is represented by the region inside circles  and  and outside . This is region .

Example Question #6 : Problem Solving Questions

What is the median of the following number set?

Possible Answers:

Correct answer:

Explanation:

In order to find the median, the set needs to be written in numerical order:

Since  and  are both the middle numbers, taking their average will give the median of the set.

Example Question #7 : Problem Solving Questions

In a group of 30 freshman students, 10 are taking Pre-calculus, 15 are taking Biology, and 10 students are taking Algebra. 5 Students are taking both Algebra and Biology, and 7 students are taking both Biology and Pre-calculus. There is no student taking both Algebra and Pre-Calculus. If none of the students take the three classes together, how many of the students don't take any of the three classes?

Possible Answers:

Correct answer:

Explanation:

Venn

Let  be the number of students who don't take any of the three classes.

Example Question #8 : Problem Solving Questions

Set B contains all prime numbers. Set C contains all even numbers. How many numbers are common to both sets? 

Possible Answers:

Impossible to determine from the information provided

All real numbers

Correct answer:

Explanation:

Prime numbers are numbers with no other factors than themselves and one. Two is the first prime number and the only even prime number. Other examples are 5, 7, 11, etc.

Even numbers are numbers divisible by 2. Set C includes all numbers ending in 0, 2, 4, 6, or 8.

Thus, there is one number common to both sets: 2.

Example Question #9 : Problem Solving Questions

Venn_1

If universal set  refers to the set of seniors at Washington High School,  is the set of seniors enrolled in physics,  is the set of seniors enrolled in calculus, and  is the set of seniors enrolled in French IV, then the above Venn diagram reflects all of the following except:

Possible Answers:

No senior is enrolled in both French IV and physics.

No senior is enrolled in both French IV and calculus.

Every senior enrolled in calculus is also enrolled in physics.

Every senior not enrolled in physics is also not enrolled in calculus.

Every senior enrolled in physics is also enrolled in calculus.

Correct answer:

Every senior enrolled in physics is also enrolled in calculus.

Explanation:

The sets  and  do not intersect, so no senior is enrolled in both French IV and physics; the sets  and  do not intersect, so no senior is enrolled in both French IV and calculus. 

, so every senior enrolled in calculus is also enrolled in physics; contrapositively, every senior not enrolled in physics is also not enrolled in calculus.

The correct choice is the remaining statement - every senior enrolled in physics is also enrolled in calculus - since  is not a subset of .

Example Question #3 : Sets

Choose the statement that is the logical opposite of:

"John is a Toastmaster but not an Elk."

Possible Answers:

John is an Elk but not a Toastmaster.

If John is not an Elk, then he is not a Toastmaster.

John is neither a Toastmaster nor an Elk.

If John is not a Toastmaster, then he is an Elk.

John is a Toastmaster and an Elk.

Correct answer:

If John is not an Elk, then he is not a Toastmaster.

Explanation:

Let  and  be the set of all Toastmasters and Elks, respectively, and let  be the set of all people.  and , so the set to which John belongs is the shaded set in this Venn diagram:

Venn_1

the logical opposite of this is that John belongs to the shaded set in the diagram:

Venn_1

A way of saying this is   or , or, equivalently, if , then .

In plain English, if John is not an Elk, then John is not a Toastmaster.

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