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 R1={(x,y)x,yR,x+y>2}R_1 = \{(x, y) \mid x, y \in \mathbb{R}, x + y > 2 \}

This relation defines pairs (x,y)(x, y) such that the sum of xx and yy is greater than 2.

  • Injective Test: A relation is injective if each value of xx corresponds to exactly one value of yy.
  • For this relation, there are many values of yy for a given xx. For example, for x=1x = 1, we can have y=2y = 2 (since 1+2=3>21 + 2 = 3 > 2) or y=3y = 3 (since 1+3=4>21 + 3 = 4 > 2). Thus, the relation is not injective.

2. Relation R2={(x,y)x,yR,x>y}R_2 = \{(x, y) \mid x, y \in \mathbb{R}, x > y \}

This relation defines pairs (x,y)(x, y) such that xx is strictly greater than yy.

  • Injective Test: A relation is injective if each value of xx corresponds to exactly one value of yy.
  • In this case, for a given xx, there are infinitely many values of yy such that x>yx > y. For example, for x=3x = 3, yy could be any real number less than 3 (e.g., 2,1,0,12, 1, 0, -1, etc.). Hence, this relation is not injective.

3. Relation R3={(x,y)x,yR,x+y=12}R_3 = \{(x, y) \mid x, y \in \mathbb{R}, x + y = 12 \}

This relation defines pairs (x,y)(x, y) such that the sum of xx and yy is always 12.

  • Injective Test: A relation is injective if each value of xx corresponds to exactly one value of yy.
  • For any value of xx, yy is determined uniquely by y=12xy = 12 - x. Hence, this relation is injective because each xx gives a unique yy.

4. Relation R4={(x,y)x,yR,y=x2}R_4 = \{(x, y) \mid x, y \in \mathbb{R}, y = x^2 \}

This relation defines pairs (x,y)(x, y) where y=x2y = x^2.

  • Injective Test: A relation is injective if each value of xx corresponds to exactly one value of yy.
  • Here, for x=2x = 2, y=4y = 4, and for x=2x = -2, y=4y = 4 as well. Hence, multiple values of xx (both 22 and 2-2) give the same value of yy. Therefore, this relation is not injective.

Conclusion:

  • R1R_1 is not injective.
  • R2R_2 is not injective.
  • R3R_3 is injective.
  • R4R_4 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:

  1. What is the difference between a one-to-one function and a many-to-one function?
  2. How can we prove that a function is not injective using examples?
  3. What are the general conditions that must be satisfied for a relation to be injective?
  4. How can we determine if a relation is surjective or bijective?
  5. 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 xx maps to more than one value of yy. 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