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Let [tex][tex]$U=\{1,2,3,4,5,6,7,8,9,10\}$[/tex][/tex] be a universal set and define the following:
[tex]\[ A=\{x \in U \mid x \ \textless \ 5\} \text{ and } B=\{x \in U \mid x \ \textgreater \ 3\} \][/tex]

Find [tex][tex]$A \cap B$[/tex][/tex] and [tex][tex]$A \cup B$[/tex][/tex]. Present your answer by listing the elements of the sets.

[tex]\[
\begin{array}{l}
A \cap B = \{\square\} \\
A \cup B = \{\square\}
\end{array}
\][/tex]



Answer :

GinnyAnswer
Let's start by defining the required sets.

1. The universal set [tex]\( U \)[/tex]:
[tex]\[ U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \][/tex]

2. Define set [tex]\( A \)[/tex] as:
[tex]\[ A = \{x \in U \mid x < 5 \} \][/tex]
This set includes all elements of [tex]\( U \)[/tex] that are less than 5:
[tex]\[ A = \{1, 2, 3, 4\} \][/tex]

3. Define set [tex]\( B \)[/tex] as:
[tex]\[ B = \{x \in U \mid x > 3 \} \][/tex]
This set includes all elements of [tex]\( U \)[/tex] that are greater than 3:
[tex]\[ B = \{4, 5, 6, 7, 8, 9, 10\} \][/tex]

Next, we need to find the intersection of [tex]\( A \)[/tex] and [tex]\( B \)[/tex], denoted as [tex]\( A \cap B \)[/tex]. The intersection of two sets is the set of elements that are common to both sets.

[tex]\[ A = \{1, 2, 3, 4\} \][/tex]
[tex]\[ B = \{4, 5, 6, 7, 8, 9, 10\} \][/tex]

The common element between [tex]\( A \)[/tex] and [tex]\( B \)[/tex] is:
[tex]\[ A \cap B = \{4\} \][/tex]

Now, we need to find the union of [tex]\( A \)[/tex] and [tex]\( B \)[/tex], denoted as [tex]\( A \cup B \)[/tex]. The union of two sets is the set of all elements that are in either set.

[tex]\[ A = \{1, 2, 3, 4\} \][/tex]
[tex]\[ B = \{4, 5, 6, 7, 8, 9, 10\} \][/tex]

Combining all elements from both sets without repeating any element, we get:
[tex]\[ A \cup B = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \][/tex]

Therefore, the final answers are:
[tex]\[ \begin{array}{l} A \cap B = \{4\} \\ A \cup B = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \end{array} \][/tex]

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