Junior Cycle Mathematics Number Systems: The sets $U$, $A$, and $B$ are d

The sets $U$, $A$, and $B$ are defined as follows, where $U$ is the universal set: $U = \{2, 3, 4, 5, \ldots, 30\}$ $A = \{\text{multiples of } 2\}$ $B = \{\text{multiples of } 3\}$ $C = \{\text{multiples of } 5\}$ - Junior Cycle Mathematics - Question 2 - 2018

Question 2

$The-sets-$U$,-$A$,-and-$B$-are-defined-as-follows,-where-$U$-is-the-universal-set:--$U-=-\{2,-3,-4,-5,-\ldots,-30\}$-$A-=-\{\text{multiples-of-}-2\}$-$B-=-\{\text{multiples-of-}-3\}$-$C-=-\{\text{multiples-of-}-5\}$-Junior Cycle Mathematics-Question 2-2018.png$

The sets $U$, $A$, and $B$ are defined as follows, where $U$ is the universal set: $U = \{2, 3, 4, 5, \ldots, 30\}$ $A = \{\text{multiples of } 2\}$ $B = \{\text{mu... show full transcript

Worked Solution & Example Answer:The sets $U$, $A$, and $B$ are defined as follows, where $U$ is the universal set: $U = \{2, 3, 4, 5, \ldots, 30\}$ $A = \{\text{multiples of } 2\}$ $B = \{\text{multiples of } 3\}$ $C = \{\text{multiples of } 5\}$ - Junior Cycle Mathematics - Question 2 - 2018

Step 1

Find $\#\left( A \cup B \cup C \right)^{'}$, the number of elements in the complement of the set $A \cup B \cup C$

96%

114 rated

Answer

To find the complement of the set $A \cup B \cup C$ , we first identify the elements in the union of sets $A$ , $B$ , and $C$ , which includes all multiples of 2, 3, and 5 in the universal set $U$ .

The multiples of 2 from 2 to 30 are: {2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30}. The multiples of 3 are: {3, 6, 9, 12, 15, 18, 21, 24, 27, 30}. The multiples of 5 are: {5, 10, 15, 20, 25, 30}.

Now, the union $A \cup B \cup C$ contains the numbers: {2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30}.

The numbers from the universal set $U$ that are not in this union are: {7, 11, 13, 17, 19, 23, 29}. Hence, the number of elements in the complement $\#\left( A \cup B \cup C \right)^{'} = 7$ .

Step 2

How many divisors does each of the numbers in $(A \cup B \cup C)^{'}$ have?

99%

104 rated

Answer

Each of the numbers in the set $\{7, 11, 13, 17, 19, 23, 29\}$ is prime. By definition, prime numbers have exactly two divisors: themselves and 1.

Step 3

What name is given to numbers that have exactly this many divisors?

96%

101 rated

Answer

These numbers are called prime numbers.

The sets $U$, $A$, and $B$ are defined as follows, where $U$ is the universal set: $U = \{2, 3, 4, 5, \ldots, 30\}$ $A = \{\text{multiples of } 2\}$ $B = \{\text{multiples of } 3\}$ $C = \{\text{multiples of } 5\}$ - Junior Cycle Mathematics - Question 2 - 2018

Worked Solution & Example Answer:The sets $U$, $A$, and $B$ are defined as follows, where $U$ is the universal set: $U = \{2, 3, 4, 5, \ldots, 30\}$ $A = \{\text{multiples of } 2\}$ $B = \{\text{multiples of } 3\}$ $C = \{\text{multiples of } 5\}$ - Junior Cycle Mathematics - Question 2 - 2018

Find $\#\left( A \cup B \cup C \right)^{'}$, the number of elements in the complement of the set $A \cup B \cup C$

How many divisors does each of the numbers in $(A \cup B \cup C)^{'}$ have?

What name is given to numbers that have exactly this many divisors?

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