Problem Analysis:
The task involves analyzing the relationships between elements in two sets, as depicted in the diagrams. Each diagram shows two sets (circles) with labeled elements and arrows indicating relationships between these elements. The goal is to determine the nature of these relationships and classify them based on their properties.
#### Key Concepts:
1.
Function: A mapping from one set to another where each element in the first set (domain) is associated with exactly one element in the second set (codomain).
2.
One-to-One Function: A function where each element in the domain maps to a unique element in the codomain, and no two elements in the domain map to the same element in the codomain.
3.
Onto Function: A function where every element in the codomain is mapped to by at least one element in the domain.
4.
Bijection: A function that is both one-to-one and onto.
Diagram Analysis:
#### (a)
-
Sets:
- Left set: {a, b, c}
- Right set: {d, e, f}
-
Arrows:
- a → d
- b → e
- c → f
-
Analysis:
- Each element in the left set maps to exactly one element in the right set.
- No two elements in the left set map to the same element in the right set.
- Every element in the right set is mapped to by exactly one element in the left set.
-
Conclusion: This is a
bijection (one-to-one and onto).
#### (b)
-
Sets:
- Left set: {a, b, c}
- Right set: {d, e, f}
-
Arrows:
- a → d
- b → e
- c → e (Note: c and b both map to e)
-
Analysis:
- Each element in the left set maps to exactly one element in the right set.
- However, two elements in the left set (b and c) map to the same element in the right set (e).
- Not all elements in the right set are mapped to (f is not mapped to).
-
Conclusion: This is a
function, but it is
neither one-to-one nor onto.
#### (c)
-
Sets:
- Left set: {a, b, c, d}
- Right set: {p, q, r, s}
-
Arrows:
- a → p
- b → q
- c → r
- d → s
-
Analysis:
- Each element in the left set maps to exactly one element in the right set.
- No two elements in the left set map to the same element in the right set.
- Every element in the right set is mapped to by exactly one element in the left set.
-
Conclusion: This is a
bijection (one-to-one and onto).
#### (d)
-
Sets:
- Left set: {a, b, c, d}
- Right set: {p, q, r}
-
Arrows:
- a → p
- b → q
- c → r
- d → r (Note: c and d both map to r)
-
Analysis:
- Each element in the left set maps to exactly one element in the right set.
- However, two elements in the left set (c and d) map to the same element in the right set (r).
- Every element in the right set is mapped to by at least one element in the left set.
-
Conclusion: This is a
function that is
onto but
not one-to-one.
Final Answer:
- (a):
Bijection
- (b):
Function (neither one-to-one nor onto)
- (c):
Bijection
- (d):
Function (onto but not one-to-one)
$$
\boxed{\text{(a): Bijection, (b): Function (neither one-to-one nor onto), (c): Bijection, (d): Function (onto but not one-to-one)}}
$$
Parent Tip: Review the logic above to help your child master the concept of function or not a function worksheet.
