QuestionDraw Venn diagrams for sets $A$ and $B$ to show that $A = (A \cap B) \cup (A \cap \bar{B})$ .

Studdy Solution

STEP 1

Assumptions1. $A$ and $B$ are two sets. . $\cap$ denotes the intersection of two sets.
3. $\cup$ denotes the union of two sets.
4. $\bar{B}$ denotes the complement of set $B$ .

STEP 2

First, let's understand the expression $A=(A \cap B) \cup(A \cap \bar{B})$ .
This expression states that set $A$ can be expressed as the union of the intersection of sets $A$ and $B$ and the intersection of sets $A$ and the complement of $B$ .

STEP 3

Draw two overlapping circles to represent sets $A$ and $B$ in a Venn diagram. Label the circles as $A$ and $B$ respectively.

STEP 4

The intersection of sets $A$ and $B$ ( $A \cap B$ ) is represented by the area where the two circles overlap.

STEP 5

The complement of set $B$ ( $\bar{B}$ ) is represented by the area outside circle $B$ but within the universal set.

STEP 6

The intersection of sets $A$ and the complement of $B$ ( $A \cap \bar{B}$ ) is represented by the area within circle $A$ but outside circle $B$ .

STEP 7

The union of the intersection of sets $A$ and $B$ and the intersection of sets $A$ and the complement of $B$ (( $A \cap B) \cup(A \cap \bar{B})$ ) is represented by the entire area within circle $A$ .

STEP 8

From the Venn diagram, it is clear that the entire area within circle $A$ is equal to the union of the intersection of sets $A$ and $B$ and the intersection of sets $A$ and the complement of $B$ . Hence, the Venn diagram verifies the given expression $A=(A \cap B) \cup(A \cap \bar{B})$ .

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QuestionDraw Venn diagrams for sets $A$ and $B$ to show that $A = (A \cap B) \cup (A \cap \bar{B})$ .

Studdy Solution

STEP 1

Assumptions1. $A$ and $B$ are two sets. . $\cap$ denotes the intersection of two sets.
3. $\cup$ denotes the union of two sets.
4. $\bar{B}$ denotes the complement of set $B$ .

STEP 2

First, let's understand the expression $A=(A \cap B) \cup(A \cap \bar{B})$ .
This expression states that set $A$ can be expressed as the union of the intersection of sets $A$ and $B$ and the intersection of sets $A$ and the complement of $B$ .

STEP 3

Draw two overlapping circles to represent sets $A$ and $B$ in a Venn diagram. Label the circles as $A$ and $B$ respectively.

STEP 4

The intersection of sets $A$ and $B$ ( $A \cap B$ ) is represented by the area where the two circles overlap.

STEP 5

The complement of set $B$ ( $\bar{B}$ ) is represented by the area outside circle $B$ but within the universal set.

STEP 6

The intersection of sets $A$ and the complement of $B$ ( $A \cap \bar{B}$ ) is represented by the area within circle $A$ but outside circle $B$ .

STEP 7

The union of the intersection of sets $A$ and $B$ and the intersection of sets $A$ and the complement of $B$ (( $A \cap B) \cup(A \cap \bar{B})$ ) is represented by the entire area within circle $A$ .

STEP 8

From the Venn diagram, it is clear that the entire area within circle $A$ is equal to the union of the intersection of sets $A$ and $B$ and the intersection of sets $A$ and the complement of $B$ . Hence, the Venn diagram verifies the given expression $A=(A \cap B) \cup(A \cap \bar{B})$ .

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QuestionDraw Venn diagrams for sets AAA and BBB to show that A=(A∩B)∪(A∩Bˉ)A = (A \cap B) \cup (A \cap \bar{B})A=(A∩B)∪(A∩Bˉ).

Studdy Solution

STEP 1

Assumptions1. AAA and BBB are two sets. . ∩\cap∩ denotes the intersection of two sets.3. ∪\cup∪ denotes the union of two sets.4. Bˉ\bar{B}Bˉ denotes the complement of set BBB.

STEP 2

First, let's understand the expression A=(A∩B)∪(A∩Bˉ)A=(A \cap B) \cup(A \cap \bar{B})A=(A∩B)∪(A∩Bˉ).This expression states that set AAA can be expressed as the union of the intersection of sets AAA and BBB and the intersection of sets AAA and the complement of BBB.

STEP 3

Draw two overlapping circles to represent sets AAA and BBB in a Venn diagram. Label the circles as AAA and BBB respectively.

STEP 4

The intersection of sets AAA and BBB (A∩BA \cap BA∩B) is represented by the area where the two circles overlap.

STEP 5

The complement of set BBB (Bˉ\bar{B}Bˉ) is represented by the area outside circle BBB but within the universal set.

STEP 6

The intersection of sets AAA and the complement of BBB (A∩BˉA \cap \bar{B}A∩Bˉ) is represented by the area within circle AAA but outside circle BBB.

STEP 7

The union of the intersection of sets AAA and BBB and the intersection of sets AAA and the complement of BBB ((A∩B)∪(A∩Bˉ)A \cap B) \cup(A \cap \bar{B})A∩B)∪(A∩Bˉ)) is represented by the entire area within circle AAA.

STEP 8

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QuestionDraw Venn diagrams for sets $A$ and $B$ to show that $A = (A \cap B) \cup (A \cap \bar{B})$ .

Assumptions1. $A$ and $B$ are two sets. . $\cap$ denotes the intersection of two sets.
3. $\cup$ denotes the union of two sets.
4. $\bar{B}$ denotes the complement of set $B$ .

First, let's understand the expression $A=(A \cap B) \cup(A \cap \bar{B})$ .
This expression states that set $A$ can be expressed as the union of the intersection of sets $A$ and $B$ and the intersection of sets $A$ and the complement of $B$ .

Draw two overlapping circles to represent sets $A$ and $B$ in a Venn diagram. Label the circles as $A$ and $B$ respectively.

The intersection of sets $A$ and $B$ ( $A \cap B$ ) is represented by the area where the two circles overlap.

The complement of set $B$ ( $\bar{B}$ ) is represented by the area outside circle $B$ but within the universal set.

The intersection of sets $A$ and the complement of $B$ ( $A \cap \bar{B}$ ) is represented by the area within circle $A$ but outside circle $B$ .

The union of the intersection of sets $A$ and $B$ and the intersection of sets $A$ and the complement of $B$ (( $A \cap B) \cup(A \cap \bar{B})$ ) is represented by the entire area within circle $A$ .