Real Analysis - GRE Quantitative Reasoning

Card 0 of 128

Question

$U=\left { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \right }$

$A=\left { 4, 9 \right }$

$B=\left { 1, 3, 5, 10 \right }$

$C=\left { 7, 8, 9, 10 \right }$

$D=\left { 1, 3, 5, 10 \right }$

What is the value of ?

Answer

When you see a letter with the prime symbol, that means complement. Complement tells you to take whatever is NOT in that set and what is in the universal set. You could also say you are eliminating the numbers in the universal set that are in the set you are complementing.

The universal set is $U=\left { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \right }$

Set $C=\left { 7, 8, 9, 10 \right }$

Taking the numbers 7, 8, 9, and 10 out of the universal set you get the answer

$\left { 1, 2, 3, 4, 5, 6 \right }$

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Question

$U=\left { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\right }$

$A=\left { 3, 4, 6, 7, 8, 9\right }$

$B=\left { 1, 3, 4, 5\right }$

$C=\left { 2, 7, 8, 9, 11, 12\right }$

$D=\left { 4, 9, 11, 12\right }$

Evaluate

Answer

The prime symbol means to take the complement of set D. To find a complement, you want to only include the numbers in the universal set that do not appear in the set being complemented.

If set $D=\left { 4, 9, 11, 12\right }$

and the universal set is $U=\left { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\right }$

The numbers you want to include are everything in the universal set except 4, 9, 10, 11, and 12

This leaves the numbers

$D'=\left { 1, 2, 3, 5, 6, 7, 8, 10\right }$

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Question

$U=\left { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\right }$

$A=\left { 3, 4, 6, 7, 8, 9\right }$

$B=\left { 1, 3, 4, 5\right }$

$C=\left { 2, 7, 8, 9, 11, 12\right }$

$D=\left { 4, 9, 11, 12\right }$

Evaluate

Answer

A complement symbol means you want to only include the numbers in the universal set that do not appear in the set being complemented.

However, all the numbers in the universal set appear in itself. This means the complement of ANY universal set is the empty set.

$\left { \right }$

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Question

$U=\left { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\right }$

$A=\left { 3, 4, 6, 7, 8, 9\right }$

$B=\left { 1, 3, 4, 5\right }$

$C=\left { 2, 7, 8, 9, 11, 12\right }$

$D=\left { 4, 9, 11, 12\right }$

Evaluate $\left ( A\cup B\right )'$

Answer

$\left ( A \cup B\right )'$

Always work in parentheses first. First, you need to solve the union of sets A and B. Union means to include everything in both sets without repeating any duplicates.

$A\cup B=\left { 1, 3, 4, 5, 6, 7, 8, 9\right }$

Now take the complement of this by including the numbers in the universal set that do not appear in A union B. This leaves

$(A \cup B)'=\left { 2, 10, 11, 12\right }$

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Question

If angle $A=35.578^{\circ}$ , find angle if angles and are complementary.

Answer

Step 1: Recall the definition of Complementary angles.

Two angles are complementary if the sum of the two angles is ALWAYS $90^{\circ}$ .

Step 2: If the sum of the angles is $90^{\circ}$ , then we can subtract the other one to find the missing angle...

So, $B=(90-35.578)^{\circ}=54.422^{\circ}$

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Question

Use DeMorgan's law to write a statement that is equivalent to the following statement

$(A \cap B')'$

Answer

The easiest way to remember DeMorgan's law is that you flip the symbol upside down (which changes union to intersection and vice versa), complement both sets (remembering that the complement of a complement is just that set), and either remove parentheses or add parentheses and place the complement symbol outside of it.

$(A \cap B')'$

Flip the symbol from an intersection to a union, complement both sets, and remove the parentheses.

$A' \cup B$ is equivalent to the original statement

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Question

Using DeMorgan's law, which of the following is equivalent to the statement

$A' \cap B$

Answer

The easiest way to remember DeMorgan's law is that you flip the symbol upside down (which changes union to intersection and vice versa), complement both sets (remembering that the complement of a complement is just that set), and either remove parentheses or add parentheses and place the complement symbol outside of it

$A' \cap B$

Flip the intersection symbol to a union, complement both sets, and add parentheses with the complement symbol outside of it.

$(A \cup B')'$

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Question

Using DeMorgan's law, is the statement $B \cup C$ equivalent to $(B' \cap C')'$ ? If not, choose the correct statement that is equivalent.

Answer

The easiest way to remember DeMorgan's law is that you flip the symbol upside down (which changes union to intersection and vice versa), complement both sets (remembering that the complement of a complement is just that set), and either remove parentheses or add parentheses and place the complement symbol outside of it.

$B \cup C$

Flip the union symbol to an intersection, complement both sets, add parentheses and a complement symbol outside the parentheses.

$(B' \cap C')'$

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Question

Using DeMorgan's law, is the statement $(A' \cup C)'$ equivalent to $A \cap C$ ? If not, choose the correct statement that is equivalent.

Answer

The easiest way to remember DeMorgan's law is that you flip the symbol upside down (which changes union to intersection and vice versa), complement both sets (remembering that the complement of a complement is just that set), and either remove parentheses or add parentheses and place the complement symbol outside of it.

$(A' \cup C)'$

Flip the union symbol to an intersection symbol, complement both sets, and remove the parentheses

$A \cap C'$

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Question

Using DeMorgan's law, which of the following is equivalent to the statement

$C \cap (B \cup A')'$

Answer

$C \cap (B \cup A')'$

The first step is to realize that the C and intersection symbol in the original question are distractions and have nothing to do with applying DeMorgan's Law.

In the parentheses, flip the symbol from an intersection to a union and complement both sets.

$C \cap (B' \cap A)$

Normally, with only two sets, you should eliminate the parentheses that was there to show that the complement symbol applied to the entire parentheses. However, with three sets you still need to know the order in which to work out the problem so you should keep the parentheses around B and A.

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Question

Using DeMorgan's law, is the statement $(A' \cup B')'$ equivalent to $A \cap B'$ ? If not, choose the correct statement that is equivalent.

Answer

The easiest way to remember DeMorgan's law is that you flip the symbol upside down (which changes union to intersection and vice versa), complement both sets (remembering that the complement of a complement is just that set), and either remove parentheses or add parentheses and place the complement symbol outside of it.

$(A' \cup B')'$

Flip the union symbol to an intersection symbol, complement both sets, and remove the parentheses.

$A \cap B$

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Question

Suppose A, B, and C are statements such that C is false if exactly one of A or B is true. If C is true, which of the following is also true?

Answer

Step 1: Let's break down the logic problem...

C is false if A or B is True.

Example: C-False, A-True, B-False.

Step 2: Try to find what happens if C is true

If C is true, "or" changes to "and", and True becomes false..

So: C-True, A-False, B-False..

A and B must both be false for C to be true..

Answer: Both A and B are False.

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Question

Which of the following sets is not an infinite set??

Answer

Step 1: Determine the difference between Infinite and Finite Sets...

Finite Sets: A set that has very limited elements in it
Infinite Sets: Any set where I can always find another number between two given boundaries.

The Set of integers and rational numbers are infinite sets..
All numbers between and is also an infinite set because I can come up with infinite decimal numbers..

The set of whole numbers between and inclusive is a finite set because I ask for a specific type of number.. a whole number. Whole numbers cannot be decimals..

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Question

Which of the following is a finite set?

Answer

Note: The set of real numbers $(\mathbb{R})$ , natural numbers $(\mathbb{N})$ are both infinite sets.
Step 1: The set of real numbers between two numbers is also an infinite set.

Step 2: The set of whole numbers between and inclusive is finite because there are only three numbers that are represented by the set. These numbers are .

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Question

Suppose A, B, and C are statements such that C is false if exactly one of A or B is true. If C is true, which of the following is also true?

Answer

Step 1: Let's break down the logic problem...

C is false if A or B is True.

Example: C-False, A-True, B-False.

Step 2: Try to find what happens if C is true

If C is true, "or" changes to "and", and True becomes false..

So: C-True, A-False, B-False..

A and B must both be false for C to be true..

Answer: Both A and B are False.

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Question

$U=\left { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \right }$

$A=\left { 4, 9 \right }$

$B=\left { 1, 3, 5, 10 \right }$

$C=\left { 7, 8, 9, 10 \right }$

$D=\left { 1, 3, 5, 10 \right }$

What is the value of ?

Answer

When you see a letter with the prime symbol, that means complement. Complement tells you to take whatever is NOT in that set and what is in the universal set. You could also say you are eliminating the numbers in the universal set that are in the set you are complementing.

The universal set is $U=\left { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \right }$

Set $C=\left { 7, 8, 9, 10 \right }$

Taking the numbers 7, 8, 9, and 10 out of the universal set you get the answer

$\left { 1, 2, 3, 4, 5, 6 \right }$

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Question

$U=\left { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\right }$

$A=\left { 3, 4, 6, 7, 8, 9\right }$

$B=\left { 1, 3, 4, 5\right }$

$C=\left { 2, 7, 8, 9, 11, 12\right }$

$D=\left { 4, 9, 11, 12\right }$

Evaluate

Answer

The prime symbol means to take the complement of set D. To find a complement, you want to only include the numbers in the universal set that do not appear in the set being complemented.

If set $D=\left { 4, 9, 11, 12\right }$

and the universal set is $U=\left { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\right }$

The numbers you want to include are everything in the universal set except 4, 9, 10, 11, and 12

This leaves the numbers

$D'=\left { 1, 2, 3, 5, 6, 7, 8, 10\right }$

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Question

$U=\left { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\right }$

$A=\left { 3, 4, 6, 7, 8, 9\right }$

$B=\left { 1, 3, 4, 5\right }$

$C=\left { 2, 7, 8, 9, 11, 12\right }$

$D=\left { 4, 9, 11, 12\right }$

Evaluate

Answer

A complement symbol means you want to only include the numbers in the universal set that do not appear in the set being complemented.

However, all the numbers in the universal set appear in itself. This means the complement of ANY universal set is the empty set.

$\left { \right }$

Compare your answer with the correct one above

Question

$U=\left { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\right }$

$A=\left { 3, 4, 6, 7, 8, 9\right }$

$B=\left { 1, 3, 4, 5\right }$

$C=\left { 2, 7, 8, 9, 11, 12\right }$

$D=\left { 4, 9, 11, 12\right }$

Evaluate $\left ( A\cup B\right )'$

Answer

$\left ( A \cup B\right )'$

Always work in parentheses first. First, you need to solve the union of sets A and B. Union means to include everything in both sets without repeating any duplicates.

$A\cup B=\left { 1, 3, 4, 5, 6, 7, 8, 9\right }$

Now take the complement of this by including the numbers in the universal set that do not appear in A union B. This leaves

$(A \cup B)'=\left { 2, 10, 11, 12\right }$

Compare your answer with the correct one above

Question

If angle $A=35.578^{\circ}$ , find angle if angles and are complementary.

Answer

Step 1: Recall the definition of Complementary angles.

Two angles are complementary if the sum of the two angles is ALWAYS $90^{\circ}$ .

Step 2: If the sum of the angles is $90^{\circ}$ , then we can subtract the other one to find the missing angle...

So, $B=(90-35.578)^{\circ}=54.422^{\circ}$

Compare your answer with the correct one above

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