Use ⊆, ⊈, ⊂, or both ⊂ and ⊆ to make a true statement. {x | x ∈ N and x > 9} {x | x ∈ N and 4 < x ≤ 9}
Answers
Step-by-step explanation:
Before beginning this section, it would be a good idea to review sets and set notation, including the roster method and set builder notation, in Section 2.3.
PREVIEW ACTIVITY 5.1.1 : Set Operations
In Section 2.1, we used logical operators (conjunction, disjunction, negation) to form new statements from existing statements. In a similar manner, there are several ways to create new sets from sets that have already been defined. In fact, we will form these new sets using the logical operators of conjunction (and), disjunction (or), and negation (not). For example, if the universal set is the set of natural numbers N and
A={1,2,3,4,5,6} and B={1,3,5,7,9},(5.1.1)
The set consisting of all natural numbers that are in A and are in B is the set {1,3,5} ;
The set consisting of all natural numbers that are in A or are in B is the set {1,2,3,4,5,6,7,9} ; and
The set consisting of all natural numbers that are in A and are not in B is the set {2,4,6}.
These sets are examples of some of the most common set operations, which are given in the following definitions.
Definition: intersection
Let A and B be subsets of some universal set U . The intersection of A and B , written A∩B and read “ A intersect B ,” is the set of all elements that are in both A and B . That is,
A∩B={x∈U|x∈A and x∈B}.(5.1.2)
The union of A and B , written A∪B and read “ A union B ,” is the set of all elements that are in A or in B . That is,
A∪B={x∈U|x∈A or x∈B}.(5.1.3)
Definition: complement
Let A and B be subsets of some universal set U . The set difference of A and B , or relative complement of B with respect to A , written A−B and read “ A minus B ” or “the complement of B with respect to A ,” is the set of all elements in A that are not in B . That is,
A−B={x∈U|x∈A and x∉B}.(5.1.4)
The complement of the set A , written Ac and read “the complement of A ,” is the set of all elements of U that are not in A . That is,
Ac={x∈U|x∉A}.(5.1.5)
For the rest of this preview activity, the universal set is U={0,1,2,3,...,10} , and we will use the following subsets of U :
A={0,1,2,3,9} and B={2,3,4,5,6},(5.1.6)
So in this case, A∩B={x∈U|x∈A and x∈B}={2,3}. Use the roster method to specify each of the following subsets of U .
A∪B
Ac
Bc
We can now use these sets to form even more sets. For example,
A∩Bc={0,1,2,3,9}∩{0,1,7,8,9,10}={0,1,9}.(5.1.7)
Use the roster method to specify each of the following subsets of U .
A∪Bc
Ac∩Bc
Ac∪Bc
(A∩B)c
Preview Activity 5.1.2 : Venn Diagrams for Two Sets
In Preview Activity 5.1.1 , we worked with verbal and symbolic definitions of set operations. However, it is also helpful to have a visual representation of sets. Venn diagrams are used to represent sets by circles (or some other closed geometric shape) drawn inside a rectangle. The points inside the rectangle represent the universal set U , and the elements of a set are represented by the points inside the circle that represents the set. For example, Figure 5.1.1 is a Venn diagram showing two sets.
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