Math Problem Statement
Which of the following relations is/are one-one function?
R 1
{ ( x , y ) ∣ x , y ∈ R , x + y
2 } R 1 ={(x,y)∣x,y∈R,x+y>2}
R 2
{ ( x , y ) ∣ x , y ∈ R , x
y } R 2 ={(x,y)∣x,y∈R,x>y}
R 3
{ ( x , y ) ∣ x + y
12 } R 3 ={(x,y)∣x+y=12}
R 4
{ ( x , y ) ∣ x , y ∈ R , y
x 2 } R 4 ={(x,y)∣x,y∈R,y=x 2 }
Solution
To determine which of the given relations are one-one functions (injective), let's break down each relation one by one.
1. Relation
This relation defines pairs such that the sum of and is greater than 2.
- Injective Test: A relation is injective if each value of corresponds to exactly one value of .
- For this relation, there are many values of for a given . For example, for , we can have (since ) or (since ). Thus, the relation is not injective.
2. Relation
This relation defines pairs such that is strictly greater than .
- Injective Test: A relation is injective if each value of corresponds to exactly one value of .
- In this case, for a given , there are infinitely many values of such that . For example, for , could be any real number less than 3 (e.g., , etc.). Hence, this relation is not injective.
3. Relation
This relation defines pairs such that the sum of and is always 12.
- Injective Test: A relation is injective if each value of corresponds to exactly one value of .
- For any value of , is determined uniquely by . Hence, this relation is injective because each gives a unique .
4. Relation
This relation defines pairs where .
- Injective Test: A relation is injective if each value of corresponds to exactly one value of .
- Here, for , , and for , as well. Hence, multiple values of (both and ) give the same value of . Therefore, this relation is not injective.
Conclusion:
- is not injective.
- is not injective.
- is injective.
- is not injective.
Would you like to dive deeper into any specific relation, or have any questions? Here are 5 related questions to expand your understanding:
- What is the difference between a one-to-one function and a many-to-one function?
- How can we prove that a function is not injective using examples?
- What are the general conditions that must be satisfied for a relation to be injective?
- How can we determine if a relation is surjective or bijective?
- How does the domain and range affect the injectivity of a relation?
Tip: When checking if a relation is injective, try to determine if a single value of maps to more than one value of . If it does, the relation is not injective.
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Math Problem Analysis
Mathematical Concepts
Relations
Functions
Injective Functions
Real Numbers
Formulas
x + y > 2
x > y
x + y = 12
y = x^2
Theorems
Injective (One-to-One) Function Test
Suitable Grade Level
Grades 11-12
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