Math / Discrete

QuestionEsercizio Siano $A=\{3,4,7,9\}$ e $B=\{0,2,4,6,8,10,12\}$ insiemi dentro all'insieme ambiente $S=\{0,1,2,3,4,5,6,7,8,9,10,11,12\}$ . Lista gli elementi di ciascuno dei seguenti insiemi. Il complemento di $A$ è denotato $A^{\prime}$ e il complemento di $B$ è denotato $B^{\prime}$ . (a) $A^{\prime}=$ $\square$ (b) $B^{\prime}=$ $\square$ (c) $A \cup B=$ $\square$ (d) $\left(A^{\prime} \cap B\right)^{\prime}=$ $\square$ (e) $\left(B^{\prime} \cup A\right)^{\prime}=$ $\square$

Studdy Solution

STEP 1

1. The universal set is $S = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\}$ .
2. The set $A = \{3, 4, 7, 9\}$ and the set $B = \{0, 2, 4, 6, 8, 10, 12\}$ .
3. The complement of a set $A$ , denoted $A^{\prime}$ , is the set of elements in $S$ that are not in $A$ .

STEP 2

1. Find the complement of set $A$ .
2. Find the complement of set $B$ .
3. Find the union of sets $A$ and $B$ .
4. Find the complement of the intersection of $A^{\prime}$ and $B$ .
5. Find the complement of the union of $B^{\prime}$ and $A$ .

STEP 3

To find the complement of $A$ , list all elements in $S$ that are not in $A$ .
$A = \{3, 4, 7, 9\}$ $A^{\prime} = S - A = \{0, 1, 2, 5, 6, 8, 10, 11, 12\}$

STEP 4

To find the complement of $B$ , list all elements in $S$ that are not in $B$ .
$B = \{0, 2, 4, 6, 8, 10, 12\}$ $B^{\prime} = S - B = \{1, 3, 5, 7, 9, 11\}$

STEP 5

To find the union of $A$ and $B$ , list all elements that are in either $A$ or $B$ or both.
$A \cup B = \{3, 4, 7, 9\} \cup \{0, 2, 4, 6, 8, 10, 12\}$ $A \cup B = \{0, 2, 3, 4, 6, 7, 8, 9, 10, 12\}$

STEP 6

First, find the intersection of $A^{\prime}$ and $B$ , then find the complement of that intersection.
$A^{\prime} \cap B = \{0, 1, 2, 5, 6, 8, 10, 11, 12\} \cap \{0, 2, 4, 6, 8, 10, 12\}$ $A^{\prime} \cap B = \{0, 2, 6, 8, 10, 12\}$
Now, find the complement:
$\left(A^{\prime} \cap B\right)^{\prime} = S - \{0, 2, 6, 8, 10, 12\}$ $\left(A^{\prime} \cap B\right)^{\prime} = \{1, 3, 4, 5, 7, 9, 11\}$

STEP 7

First, find the union of $B^{\prime}$ and $A$ , then find the complement of that union.
$B^{\prime} \cup A = \{1, 3, 5, 7, 9, 11\} \cup \{3, 4, 7, 9\}$ $B^{\prime} \cup A = \{1, 3, 4, 5, 7, 9, 11\}$
Now, find the complement:
$\left(B^{\prime} \cup A\right)^{\prime} = S - \{1, 3, 4, 5, 7, 9, 11\}$ $\left(B^{\prime} \cup A\right)^{\prime} = \{0, 2, 6, 8, 10, 12\}$
The solutions are: (a) $A^{\prime} = \{0, 1, 2, 5, 6, 8, 10, 11, 12\}$ (b) $B^{\prime} = \{1, 3, 5, 7, 9, 11\}$ (c) $A \cup B = \{0, 2, 3, 4, 6, 7, 8, 9, 10, 12\}$ (d) $\left(A^{\prime} \cap B\right)^{\prime} = \{1, 3, 4, 5, 7, 9, 11\}$ (e) $\left(B^{\prime} \cup A\right)^{\prime} = \{0, 2, 6, 8, 10, 12\}$

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Studdy Solution

STEP 1

1. The universal set is $S = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\}$ .
2. The set $A = \{3, 4, 7, 9\}$ and the set $B = \{0, 2, 4, 6, 8, 10, 12\}$ .
3. The complement of a set $A$ , denoted $A^{\prime}$ , is the set of elements in $S$ that are not in $A$ .

STEP 2

1. Find the complement of set $A$ .
2. Find the complement of set $B$ .
3. Find the union of sets $A$ and $B$ .
4. Find the complement of the intersection of $A^{\prime}$ and $B$ .
5. Find the complement of the union of $B^{\prime}$ and $A$ .

STEP 3

To find the complement of $A$ , list all elements in $S$ that are not in $A$ .
$A = \{3, 4, 7, 9\}$ $A^{\prime} = S - A = \{0, 1, 2, 5, 6, 8, 10, 11, 12\}$

STEP 4

To find the complement of $B$ , list all elements in $S$ that are not in $B$ .
$B = \{0, 2, 4, 6, 8, 10, 12\}$ $B^{\prime} = S - B = \{1, 3, 5, 7, 9, 11\}$

STEP 5

To find the union of $A$ and $B$ , list all elements that are in either $A$ or $B$ or both.
$A \cup B = \{3, 4, 7, 9\} \cup \{0, 2, 4, 6, 8, 10, 12\}$ $A \cup B = \{0, 2, 3, 4, 6, 7, 8, 9, 10, 12\}$

STEP 6

STEP 7

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Studdy Solution

STEP 1

STEP 2

1. Find the complement of set A A A.2. Find the complement of set B B B.3. Find the union of sets A A A and B B B.4. Find the complement of the intersection of A′ A^{\prime} A′ and B B B.5. Find the complement of the union of B′ B^{\prime} B′ and A A A.

STEP 3

To find the complement of A A A, list all elements in S S S that are not in A A A.A={3,4,7,9} A = \{3, 4, 7, 9\} A={3,4,7,9} A′=S−A={0,1,2,5,6,8,10,11,12} A^{\prime} = S - A = \{0, 1, 2, 5, 6, 8, 10, 11, 12\} A′=S−A={0,1,2,5,6,8,10,11,12}

STEP 4

To find the complement of B B B, list all elements in S S S that are not in B B B.B={0,2,4,6,8,10,12} B = \{0, 2, 4, 6, 8, 10, 12\} B={0,2,4,6,8,10,12} B′=S−B={1,3,5,7,9,11} B^{\prime} = S - B = \{1, 3, 5, 7, 9, 11\} B′=S−B={1,3,5,7,9,11}

STEP 5

STEP 6

STEP 7

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1. Find the complement of set $A$ .
2. Find the complement of set $B$ .
3. Find the union of sets $A$ and $B$ .
4. Find the complement of the intersection of $A^{\prime}$ and $B$ .
5. Find the complement of the union of $B^{\prime}$ and $A$ .

To find the complement of $A$ , list all elements in $S$ that are not in $A$ .
$A = \{3, 4, 7, 9\}$ $A^{\prime} = S - A = \{0, 1, 2, 5, 6, 8, 10, 11, 12\}$

To find the complement of $B$ , list all elements in $S$ that are not in $B$ .
$B = \{0, 2, 4, 6, 8, 10, 12\}$ $B^{\prime} = S - B = \{1, 3, 5, 7, 9, 11\}$