A union B Formula

The union of two sets A and B is a set that contains all the elements of A and B and is denoted by A U B (which can be read as "A or B" (or) "A union B"). A union B formula is used to find the union of two sets A and B. The union can be found by just putting all the elements of A and B in one set and removing duplicates. i.e., A U B can be found without using the A union B formula also.

Let us learn about the A union B formula with a few examples in the end.

The union of two sets is represented by writing the symbol "U" in between the two sets. The formula for A union B means that any element that is present either in A or in B is present in A ∪ B. By using the above definition, the A union B formula is,

A ∪ B = {x : x ∈ A (or) x ∈ B}

This is the Venn diagram representing "A union B" where it represents the entire portion shaded in "Orange" color.

Example: If A = {1, 2, 3, 4} and B = {3, 4, 5} then calculate A ∪ B.

Solution:

To calculate A ∪ B, just write all the elements of A and B in a set and remove duplicates. Then we get:

A ∪ B = {1, 2, 3, 4, 5}

Suppose we have two sets A and B. Now, A U B consists of elements of both sets A and B, taken one at a time (leaving duplicates). We have the following representations:

n(A) + n(B) gives the total number of elements that are in A and the elements in B including the elements that are common. This implies that the number of common elements is counted twice. Hence, to balance that and make sure that all the elements are counted just once, we subtract n(A ∩ B) from n(A) + n(B) and hence the formula for the number of elements in A U B is:

n(A U B) = n(A) + n(B) - n(A ∩ B)

The complement of a set is the difference of the universal set and the given set. In the same way, the complement of A U B (which is denoted by (A U B)') is "universal set - (A U B)". It is shown by the Venn diagram below:

Here, the complement of A union B is the portion shown by Blue colour. This can be easily calculated using one of De Morgan's laws as:

(A U B)' = A' ∩ B'

Using the same logic as above and to avoid counting the same elements twice, we have the formula for P(A U B) (the probability formula of A union B) as:

P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

If A and B are mutually exclusive events, then we have P(A ∪ B) = P(A) + P(B) as P(A ∩ B) = 0.

Here are multiple combinations of union and intersection involved with 3 given sets A, B, and C.

A Union B Union C: The formula related to A U B U C is and is 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). It is shown by the following Venn diagram in Orange colour.

A Intersection B Union C: This can mathematically be written as:A n ( B U C) and here is the Venn diagram representing it.

Important Notes on A union B Formula:

☛ Related Topics:

Example 1: If A = {1, 2, 3, 4} and B = {2, 3, 6, 7}, then find A U B.

Solution: Given that:

A = {1, 2, 3, 4}

B = {2, 3, 6, 7}

By using the A union B formula, we find A U B just by writing all the elements of A and B in one set by avoiding duplicates.

So, A U B = {1, 2, 3, 4} U {2, 3, 6, 7} = {1, 2, 3, 4, 6, 7}

Note: The elements in the answer set do not need to be in order.

Answer: A U B = {1, 2, 3, 4, 6, 7}

Example 2: Find A U B using the following figure.

Solution: By using the A union B formula, we find A U B just by writing all the elements of A and B in one set by avoiding duplicates.

Thus, by the given Venn Diagram, A U B = {11, 20, 14, 2, 10, 15, 30}.

Note: 17, 16, 3, and 18 are neither the elements of A nor the elements of B and hence they are not present in A U B.

Answer: A U B = {11, 20, 14, 2, 10, 15, 30}.

Example 3: Determine the number of elements in A union B if n(A) = 15, n(B) = 7 and n(A ∩ B) = 3.

Solution: To determine the number of elements in A U B, we will use the formula n(A U B) = n(A) + n(B) - n(A ∩ B).

n(A U B) = 15 + 7 - 3

= 22 - 3

= 19

Answer: Hence, the number of elements in A union B is 19.

Example 4: Determine the probability of randomly getting an ace or a black card from a deck of 52 playing cards.

Solution: We know that there are 26 red cards and 26 black cards in a deck of 52 playing cards and four aces in total out of which 2 are red and 2 are black.

Let A be the event of getting an ace and B be the event of getting a black card. Then, A ∩ B is the event of getting a black ace card.

P(A) = 4/52 = 1/13

P(B) = 26/52 = 1/2

P(A ∩ B) = 2/52 = 1/26

So, the probability of getting an ace or a black card is P(A U B) which is given by,

P(A) + P(B) - P(A ∩ B)

= 4/52 + 26/52 - 2/52

= 28/52

= 7/13

Answer: Hence the required probability is 7/13.

go to slidego to slidego to slidego to slide

Book a Free Trial Class

go to slidego to slide

By using the definition of A U B, the A union B formula is, A U B = {x : x ∈ A (or) x ∈ B} which implies A U B consists of elements that are either in A or in B or in both.

A U B can be calculated by taking the elements of A and the elements of B and considering the elements that are common in both A and B only once and combining them in one set.

Yes, the sets are commutative with respect to the operation "union". For any two sets A and B, we always can prove that A ∪ B = B ∪ A.

The union of two sets A and B is a set that contains all the elements of A and B taken one at a time and is denoted by A U B.

The formula for A union B whole complement formula is given by, (AUB)c = Ac ∩ Bc. It is also one of the De-Morgan's Laws of Union of Sets.

Yes, the A union B formula can be extended to find the union of more than two sets. For example, the formula for the union of three sets A, B, and C would be A ∪ B ∪ C = (A ∪ B) ∪ C.

The cardinality (number of elements) of A union B can be calculated by counting the elements in A and B and taking the elements that are common only once. The formula for the cardinality of A union B is n(A U B) = n(A) + n(B) - n(A ∩ B).