3.06 Quiz: Unions Practice Test

The union of sets A and B contains every unique element from both sets. Although the element 2 appears in both, it is listed only once, resulting in {1, 2, 3}.

The union operation merges all unique elements from the involved sets, taking every element that appears in any of the sets. This is different from the intersection, which only finds common elements.

The union of A and B includes every unique element from both sets. This means elements a, b, and c are all part of the union, with duplicates removed.

When you take the union of any set with the empty set, the empty set does not contribute any new elements. Thus, the union remains exactly the original set.

The union of A and B includes every unique number from both sets, hence the result is {1, 2, 3, 4, 5}. Even though 3 is common to both sets, it is only counted once.

The union of sets X and Y gathers all the elements that appear in either set, resulting in {x, y, z}. This includes x from X and z from Y, together with the common element y.

By combining all elements of P and Q and removing duplicates, we obtain {1, 2, 3, 4, 5, 6}. This shows that union brings together every distinct element from both input sets.

The union operation is both commutative and associative, meaning the order and grouping of the sets do not affect the outcome. This flexibility underpins many set theory operations.

This identity shows that the union distributes over the intersection by relating A ∪ (B ∩ C) with (A ∪ B) ∩ (A ∪ C). It is a fundamental property in set theory that helps simplify complex set expressions.

The overall union gathers every unique element present in any of the sets A, B, and C. Hence, by merging these sets, the resulting set is {x, y, z, w}.

By listing all elements of the involved sets and then removing any duplicates, you directly observe all unique elements in the union. This method is straightforward and minimizes errors in manual calculations.

The union of sets M and N includes every distinct element from both sets. Therefore, the combined set is {a, b, c, d, e}.

Shading both circles in a Venn diagram represents the union because it covers all elements from both sets. This visual approach clearly shows that every part of both sets is included.

Disjoint sets have no elements in common, so their union is simply the collection of all elements from both sets. This operation does not remove any elements since there are no duplicates to consider.

A union with its own complement covers every element within the universal set. This is a fundamental identity in set theory, ensuring that the result is always U.

The union over an indexed collection of sets means collecting all elements that belong to at least one of the sets. This definition is essential for working with both finite and infinite collections in set theory.

De Morgan's laws state that the complement of a union is the intersection of the complements. This identity is fundamental in set theory and logic, providing a way to simplify expressions involving complements.

The Inclusion-Exclusion Principle is essential when counting the total number of elements in the union of several overlapping sets. It corrects for overcounting by subtracting the sizes of pairwise intersections and adding back the size of the triple intersection.

Since A is a subset of B, every element of A is contained in B. When taking the union with any set C, the relationship is preserved, ensuring that A ∪ C is a subset of B ∪ C.