A Union B Union C

A union B union C is a collection of elements that belong to sets A, B, and C. This collection of elements is denoted as A U B U C and is read as 'A union B union C'. A U B U C consists of elements that are in A or B or C. The union of three sets A, B, and C can be determined by taking all elements of three sets in a single set and avoiding duplicates. We can also determine the number of elements in A union B union C using a formula that makes sure that common elements are not counted more than once.

In this article, we will explore the concept of A union B union C in detail along with its Venn diagram, formula, and proof of A U B U C formula using the formula of the union of two sets. We will also understand the A union B union C complement with the help of its Venn diagram and solved examples for a better understanding.

A union B union C is defined as the union of three sets A, B, and C which consists of elements belonging to these three sets. As we know that the union of sets is a set operation and is represented using the 'U' symbol, the union of three sets A, B, and C is denoted by A U B U C which is read as 'A union B union C'. A U B U C consists of elements that are :

• only in A;
• only in B;
• only in C;
• elements that are common in A and B;
• elements that are common in B and C;
• elements that are common in A and C and
• elements that are common in A, B, and C.

In simple words, we can that the elements in A union B union C are in either A or B or C.

Now that we have understood the meaning of A U B U C in words, let us now see the A union B union C Venn diagram to understand the concept visually. The Venn diagram is given below highlighting the orange shaded region of A U B U C and showing the portion covered by A union B union C from the universal set. As we can see, the union of sets A, B, C includes elements that are only one of these sets, elements that are common in any two of these sets, and elements that are common in all three sets in the center.

Now, we have understood that any element that is present in set A or set B or set C is present in A U B U C. Before we proceed to the formulas, let us write the symbols used to denote the number of elements in each set.

• n(A U B U C) = Number of elements in A U B U C
• n(A) = Number of elements in A
• n(B) = Number of elements in B
• n(C) = Number of elements in C
• n(A ∩ B) = Number of elements that are common to both A and B
• n(B ∩ C) = Number of elements that are common to both B and C
• n(A ∩ C) = Number of elements that are common to both A and C
• n(A ∩ B ∩ C) = Number of elements that are common in A, B and C.

We can determine the number of elements present in A union B union C using a formula. Now, n(A) + n(B) + n(C) gives the total number of elements that are in A, B and C but it counts the common elements more than once. So, to make sure that all elements are counted just once, we subtract the number of common elements in two of these sets from n(A) + n(B) + n(C) and add the number of elements that are common in all three sets to get the exact number of elements in A U B U C. Hence, using the definition of A U B U C and the given facts, we have the following A union B union C formulas:

• A U B U C = {x: x ∈ A (or) x ∈ B (or) x ∈ C}
• n(A U B U C) = n(A) + n(B) + n(C) - n(A ∩ B) - n(B ∩ C) - n(A ∩ C) + n(A ∩ B ∩ C)
• P(A U B U C) = P(A) + P(B) + P(C) - P(A ∩ B) - P(B ∩ C) - P(A ∩ C) + P(A ∩ B ∩ C)
• P(A U B U C) = P(A) + P(B) + P(C), if A, B, C are mutually exclusive.

In the last two formulas "P" stands for probability. For example, P(A) stands for "probability of event A".

A Union B Union C Formula Proof

As we know that the number of elements in A union B union C is given by the formula n(A U B U C) = n(A) + n(B) + n(C) - n(A ∩ B) - n(B ∩ C) - n(A ∩ C) + n(A ∩ B ∩ C), next we will prove this formula using the formula for the number of elements in the union of two sets, that is, n(P U Q) = n(P) + n(Q) - n(P ∩ Q). We will use the following formulas to derive the formula for n(A U B U C):

• n(P ∩ (Q U R)) = n((P ∩ Q) U (P ∩ R))
• n(P U Q) = n(P) + n(Q) - n(P ∩ Q)
• n((P ∩ Q) ∩ (P ∩ R)) = n(P ∩ Q ∩ R)

n(A U B U C) = n(A U (B U C))

= n(A) + n(B U C) - n(A ∩ (B U C))

= n(A) + [n(B) + n(C) - n(B ∩ C)] - n((A ∩ B) U (A ∩ C))

= n(A) + n(B) + n(C) - n(B ∩ C) - [n(A ∩ B) + n(A ∩ C) - n((A ∩ B) ∩ (A ∩ C))]

= n(A) + n(B) + n(C) - n(B ∩ C) - n(A ∩ B) - n(A ∩ C) + n((A ∩ B) ∩ (A ∩ C))

= n(A) + n(B) + n(C) - n(B ∩ C) - n(A ∩ B) - n(A ∩ C) + n(A ∩ B ∩ C)

Hence, we have proved the formula for the number of elements in A union B union C.

Now that we know the concept of A union B union C, next, we will understand its complement, that is, the A union B union C complement which is denoted by (A U B U C)' or (A U B U C)^c. (A U B U C)' consists of elements of the universal set which are not in any of the sets A, B, and C. In other words, we can also say that A U B U C complement is equal to the intersection of complements of the sets A, B and C, that is, (A U B U C)' = A' ∩ B' ∩ C'. The Venn diagram given below shows the shaded region in orange that indicates the complement of A U B U C.

Important Notes on A U B U C:

• The set of elements in A U B U C is written as A U B U C = {x : x ∈ A (or) x ∈ B (or) x ∈ C}
• The formula for the number of elements in A U B U C is n(A U B U C) = n(A) + n(B) + n(C) - n(A ∩ B) - n(B ∩ C) - n(A ∩ C) + n(A ∩ B ∩ C).
• A union B union C complement consists of elements of the universal set which are not in any of the sets A, B, and C.

☛ Related Topics:

• A intersection B Formula
• Finite and Infinite Sets
• Equal Sets

What is A Union B Union C in Math?

A union B union C is a collection of elements of the sets A, B, and C. It consists of elements belonging to the three sets A, B and C. Mathematically, A union B union C is denoted by A U B U C, where U represents the union of the three sets.

How to Find A Union B Union C?

We can find A union B union C by combining all the elements of the sets A, B and C in a single set and avoiding the repetition of the elements. As we know that A U B U C = {x: x ∈ A (or) x ∈ B (or) x ∈ C}, this implies A union B union C consists of elements that are in A or B or C.

What is the Formula of A Union B Union C?

The formula for A union B union C is given by, A U B U C = {x : x ∈ A (or) x ∈ B (or) x ∈ C}. It shows that the elements in A union B union C are either in A or B or C. There are two other formulas for A union B union C which tell the number of elements and the probability of A union B union C given by,

• n(A U B U C) = n(A) + n(B) + n(C) - n(A ∩ B) - n(B ∩ C) - n(A ∩ C) + n(A ∩ B ∩ C)
• P(A U B U C) = P(A) + P(B) + P(C) - P(A ∩ B) - P(B ∩ C) - P(A ∩ C) + P(A ∩ B ∩ C)

☛Note: P(A U B U C) = P(A) + P(B) + P(C), if A, B, C are mutually exclusive.

What is the Probability of A Union B Union C?

The formula for the probability of A union B union C is given by, P(A U B U C) = P(A) + P(B) + P(C) - P(A ∩ B) - P(B ∩ C) - P(A ∩ C) + P(A ∩ B ∩ C). If the sets A, B, and C are mutually exclusive then the formula becomes P(A U B U C) = P(A) + P(B) + P(C).

What are the Elements of A U B U C?

The elements of A U B U C are the elements that are in any of three sets A or B or C. These elements can be in only one of these sets A, B, C; elements that are common in any two of these sets; and elements that are common in all three sets.

What is n(A U B U C)?

n(A U B U C) gives the number of elements in A U B U C. We can determine the number of elements present in A union B union C using the formula n(A U B U C) = n(A) + n(B) + n(C) - n(A ∩ B) - n(B ∩ C) - n(A ∩ C) + n(A ∩ B ∩ C).