ISEE Upper Level Quantitative Reasoning - How to use a Venn Diagram

Venn

In the above Venn diagram, the universal set is defined as $U = \left { a, b, c, d, e, f, g, h\right }$ . Each of the eight letters is placed in its correct region. Which of the following is equal to $\overline{A\cup B}$ ?

$\overline{A\cup B} = \left { c\right }$

$\overline{A\cup B} = \left { a,c,f\right }$

$\overline{A\cup B} = \left { a,f\right }$

$\overline{A\cup B} = \left { b,c,d,e,g,h\right }$

$\overline{A\cup B} = \left { b,d,e,g,h\right }$

Explanation

$\overline{A\cup B}$ is the complement of $A\cup B$ - the set of all elements in not in $A\cup B$ .

$A\cup B$ is the union of sets and - the set of all elements in either or . Therefore, $\overline{A\cup B}$ is the set of all elements in neither nor , which can be seen from the diagram to be only one element - . Therefore,

$\overline{A\cup B} = \left { c\right }$

Venn

In the above Venn diagram, the universal set is defined as $U = \left { a, b, c, d, e, f, g, h\right }$ . Each of the eight letters is placed in its correct region.

What is $A \cup B$ ?

$A \cup B = \left { a,b,d,e,f,g,h\right }$

$A \cup B = \left { a,f \right }$

$A \cup B = \left { b,d,e,g,h\right }$

$A \cup B = \left { a,c,f \right }$

$A \cup B = \left { a,b,c, d,e,f,g,h\right }$

Explanation

$A \cup B$ is the union of sets and - that is, the set of all elements of that are elements of either or . We want all of the letters that fall in either circle, which from the diagram can be seen to be all of the letters except . Therefore,

$A \cup B = \left { a,b,d,e,f,g,h\right }$

In a group of plants, are green, have large leaves, and are both green and have large leaves. How many plants are green without large leaves?

Explanation

Based on the information, you can draw the following Venn Diagram:

Venndiagram-7

It is very easy to solve for the number of plants that have green leaves but not large ones. This is merely . We find this by eliminating the large-leaved plants from the green ones (by subtracting the overlap from the green ones).

In a group of people, have books, have pens, and have neither books nor pens. How many people in the group have only books?

Cannot be determined

Explanation

Based on the information given, you can draw the following Venn Diagram:

Venndiagram-8

Now, you must begin by solving for . You know that the two circles together will have in them. This is arrived at by subtracting the people who have neither books nor pens () from the "universe" of people in the sample space (). Now, we know that . This is because of the overlap of in both groups. We have to get rid of one instance of that. Thus we can solve for :

Now, we can find the number of people with only books by subtracting from the to get .

In a group of people, have a laptop and have a tablet. Of those people who have a laptop or a tablet, have both. How many people in the total group have neither a laptop nor a tablet?

No answer possible

Explanation

Based on the information given, you can draw the following Venn Diagram:

Venndiagram-6

To solve this, remember that the total number of values in the two circles is:

(We must do this because of the overlap. You need to subtract out one instance of that overlap.)

If we assign the value for the unknown region, we know:

Venn

In the above Venn diagram, the universal set is defined as $U = \left { a, b, c, d, e, f, g, h\right }$ . Each of the eight letters is placed in its correct region.

What is $A \cap B$ ?

$A \cap B = \left { a,f \right }$

$A \cap B = \left { a,b,d,e,f,g,h \right }$

$A \cap B = \left { b,d,e,g,h \right }$

$A \cap B = \left { b,c,d,e,g,h \right }$

$A \cap B = \left { a,c,f \right }$

Explanation

$A \cap B$ is the intersection of sets and - that is, the set of all elements of that are elements of both and . We want all of the letters that fall in both circles, which from the diagram can be seen to be and . Therefore,

$A \cap B = \left { a,f \right }$

Venn 2

Examine the above Venn diagram. Let be the universal set of the Presidents of the United States. is the set of all Presidents born in Virginia; is the set of all Presidents born after 1850; is the set of all Presidents whose first name was or is James.

James Abram Garfield was born in Ohio in 1831. In which region would he fall?

III

Explanation

Carter would not fall in set A, since he was not a President born in Virginia.

He would not fall in B, since he was born before 1850.

He would fall in C, since his first name is James.

He would fall in the region included in set C, but not A or B - this is Region V.

Venn 2

James Earl Carter was born in Georgia in 1924. In which region would he fall?

III

Explanation

Carter would not fall in set A, since he was not a President born in Virginia.

He would fall in B, since he was born after 1850.

He would fall in C, since his first name is James.

He would fall in the region included in sets B and C, but not A - this is Region III.

Venn 2

Examine the above Venn diagram. Let universal set represent the set of all words in the English language.

Let be the set of all words whose last letter is a vowel. Let be the set of all words whose first letter is a consonant. Let be the set of all words exactly six letters in length.

Which of the following would be a subset of the set represented by the shaded region in the diagram?

Note: for purposes of this question, "Y" is considered a consonant.

{plateau, portmanteau, calliope, marionette, taco}

{tomato, potato, ravine, cabana, marine}

{autistic, estrogen, ideology, opal, understand}

{autism, enough, ideals, occult, unduly}

{apnea, esoterica, irradiate, opulence, uvula}

Explanation

The subset must comprise words that fall inside sets and , but not . Therefore, all of the words in the subset must begin with a consonant, end with a vowel, and not have six letters.

Of the given choices, the only set whose elements fit this description is {plateau, portmanteau, calliope, marionette, taco}.

Venn

$\overline{A\cap B} = \left { b,c,d,e,g,h\right }$

$\overline{A\cap B} = \left { b,d,e,g,h\right }$

$\overline{A\cap B} = \left { c\right }$

$\overline{A\cap B} = \left { a,c,f\right }$

$\overline{A\cap B} = \left { a,f\right }$

Explanation

$\overline{A \cap B}$ is the complement of $A \cap B$ - the set of all elements in not in $A \cap B$ .

$A \cap B$ is the intersection of sets and - that is, the set of all elements of that are elements of both and . Therefore, $\overline{A \cap B}$ is the set of all elements that are not in both and , which can be seen from the diagram to be all elements except and . Therefore,