Question

Using a Venn diagram, show the relation between N, Z, Q, Q', and R, where the symbols have their own meanings.

Answer 1

N ⊆ Z, Q ⊆ Z, Q' ∩ Q = ∅, R ⊆ Z

1. N is entirely contained within Z.

2. Q is wholly within Z, and Q' has no common elements with Q.

3. R is completely contained within Z.

Step-by-step explanation:

In the given context, the relations between sets N, Z, Q, Q', and R are depicted through set notation and operations. The statement "N ⊆ Z" signifies that every element in set N is also an element of set Z, indicating that N is entirely contained within Z. Similarly, "Q ⊆ Z" asserts that every element in set Q is also found in set Z, illustrating that Q is wholly within Z. The expression "Q' ∩ Q = ∅" describes the intersection of the complement of Q (Q') and Q, resulting in an empty set. This implies that there are no common elements between Q and its complement, denoted as Q' — a distinctive relationship in set theory.

Lastly, "R ⊆ Z" conveys that every element in set R is also an element of Z, establishing that R is completely contained within Z. These set relations provide a structured way to understand the containment and intersection of elements among the sets.

In mathematical terms, these relationships help define the inclusion of sets within others and the absence of common elements between specific sets. Such concepts find applications in various mathematical and logical domains, contributing to problem-solving skills and analytical thinking.

Answer 2

N represents the universal set. Z is a subset of N. Q and Q' are subsets of Z, and R is a separate set outside Q and Q'.

Step-by-step explanation:

In the Venn diagram, N is the universal set encompassing all elements under consideration. Z, being a subset of N, includes elements that are part of N. Q and Q' are subsets of Z, meaning they are distinct parts of Z. Q contains certain elements of Z, while Q' contains elements from Z that are not in Q. R, however, is a separate set, not part of Q or Q'. It could intersect with N, Z, Q, or Q' or exist entirely outside these sets.

In essence, N acts as the encompassing set, Z is a subset within N, Q and Q' are subsets of Z, and R is a separate set that could have its elements distinct from those in N, Z, Q, and Q'. This representation allows us to understand the relationship between these sets in terms of containment, where each set's elements are categorized and defined within the context of the larger sets they belong to or apart from.