Use the Venn diagram to write the descriptive and roster forms of the sets below. (a) Set: G ∪ Y (b) Set: (G ∪ Y)'
Understand the Problem
The question involves understanding set theory concepts, specifically union and complement, represented visually in a Venn diagram. The sets represent train stations along different train lines (Green and Yellow). The question asks to determine the descriptive and roster forms of the union of the Green and Yellow lines (G ∪ Y) and the complement of their union (G ∪ Y)'.
Answer
(a) $G \cup Y$: The set of all train stations along either the Green Line or the Yellow Line or both; {Arbor, Belen, Dover, Flint, Huron, Ozark, Provo, Salem, Utica} (b) $(G \cup Y)'$: The set of all train stations that are neither along the Green Line nor the Yellow Line; {}
Answer for screen readers
(a) Set: $G \cup Y$ • Descriptive form: The set of all train stations along either the Green Line or the Yellow Line or both. • Roster form: {Arbor, Belen, Dover, Flint, Huron, Ozark, Provo, Salem, Utica} (b) Set: $(G \cup Y)'$ • Descriptive form: The set of all train stations that are neither along the Green Line nor the Yellow Line • Roster form: {}
Steps to Solve
- Find the descriptive form and roster form of $G \cup Y$
$G \cup Y$ represents the union of set G and set Y. The union of two sets includes all elements that are in either set G or set Y or in both. From the Venn diagram, the elements in G are Huron, Salem, Arbor, Flint, Provo, and Utica. The elements in Y are Salem, Ozark, Dover, and Belen. Therefore, $G \cup Y$ contains the elements Arbor, Flint, Provo, Utica, Huron, Salem, Ozark, Dover, and Belen. The descriptive form would be: "The set of all train stations along either the Green Line or the Yellow Line or both". The roster form would be {Arbor, Flint, Provo, Utica, Huron, Salem, Ozark, Dover, Belen}.
- Find the descriptive form and roster form of $(G \cup Y)'$
$(G \cup Y)'$ represents the complement of the union of set G and set Y. The complement of a set includes all elements in the universal set U that are not in the set. From the Venn diagram, the universal set U consists of all the listed stations: Arbor, Belen, Dover, Flint, Huron, Ozark, Provo, Salem, and Utica. The set $G \cup Y$ consists of Arbor, Flint, Provo, Utica, Huron, Salem, Ozark, Dover, and Belen. Therefore, $(G \cup Y)'$ would include any elements in U that are not in $G \cup Y$. Since $G \cup Y$ already includes all the elements listed in U, the complement is the empty set. The descriptive form would be: "The set of all train stations that are neither along the Green Line nor the Yellow Line". The roster form would be {}.
(a) Set: $G \cup Y$ • Descriptive form: The set of all train stations along either the Green Line or the Yellow Line or both. • Roster form: {Arbor, Belen, Dover, Flint, Huron, Ozark, Provo, Salem, Utica} (b) Set: $(G \cup Y)'$ • Descriptive form: The set of all train stations that are neither along the Green Line nor the Yellow Line • Roster form: {}
More Information
The union of two sets combines the elements of both sets. The complement of a set contains all elements not in the set, relative to the universal set.
Tips
A common mistake is to not carefully consider all elements in the universal set when finding the complement. Also, students sometimes confuse union and intersection.
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