Complements - GRE Subject Test: Math

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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.

$\left { \right }$

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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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