Match the following in List - I and List - II, for a function f: List – I List - II (a) ∀ x ∀ y (f (x) = f(y) → |x = y) (i) Constant (b) ∀ y ∃ x (f (x) = y) (ii) Injective (c) ∀ x f (x) = k (iii) Surjective
(a) – (ii), (b) – (iii), (c) – (i)
Understanding Function Properties and Types
This question asks us to match mathematical definitions of function properties with their corresponding types. Let's break down each definition provided in List-I and connect it to the correct type in List-II.
We are given a function $f$ defined from some domain to some codomain.
Analyzing List-I Definitions
- (a) $\forall x \forall y (f (x) = f(y) \rightarrow x = y)$
This statement reads: "For all x and for all y (in the domain), if the function values $f(x)$ and $f(y)$ are equal, then the inputs $x$ and $y$ must be equal."
In simpler terms, this means that distinct inputs always produce distinct outputs. If two inputs give the same output, those inputs must have been the same in the first place. This is the formal definition of a one-to-one function, also known as an injective function.
- (b) $\forall y \exist x (f (x) = y)$
This statement reads: "For all y (in the codomain), there exists at least one x (in the domain) such that $f(x)$ is equal to y."
In simpler terms, this means that every element in the codomain is 'hit' by the function; there is always an input in the domain that maps to any given output in the codomain. The range of the function is equal to its codomain. This is the formal definition of an onto function, also known as a surjective function.
- (c) $\forall x f (x) = k$
This statement reads: "For all x (in the domain), $f(x)$ is equal to k," where k is a fixed value.
In simpler terms, regardless of the input from the domain, the function always produces the exact same output value, $k$. This is the definition of a constant function.
Matching List-I with List-II
Based on our analysis of the definitions:
- Definition (a) corresponds to an Injective function.
- Definition (b) corresponds to a Surjective function.
- Definition (c) corresponds to a Constant function.
Let's map these:
- (a) $\rightarrow$ (ii) Injective
- (b) $\rightarrow$ (iii) Surjective
- (c) $\rightarrow$ (i) Constant
This gives us the matching (a) - (ii), (b) - (iii), (c) - (i).
Confirming the Matching
We can summarize the matching in a table:
| List-I (Definition) | List-II (Function Type) | Matching |
|---|---|---|
| (a) $\forall x \forall y (f (x) = f(y) \rightarrow x = y)$ | (i) Constant | (a) matches (ii) Injective |
| (b) $\forall y \exist x (f (x) = y)$ | (ii) Injective | (b) matches (iii) Surjective |
| (c) $\forall x f (x) = k$ | (iii) Surjective | (c) matches (i) Constant |
The correct matching is (a) - (ii), (b) - (iii), (c) - (i).
Revision Table: Function Types and Definitions
| Function Type | Definition (Formal) | Definition (Simple) |
|---|---|---|
| Injective (One-to-One) | $\forall x \forall y (f (x) = f(y) \rightarrow x = y)$ OR $\forall x \forall y (x \neq y \rightarrow f(x) \neq f(y))$ | Each element in the domain maps to a unique element in the codomain. No two distinct inputs give the same output. |
| Surjective (Onto) | $\forall y \exist x (f (x) = y)$ | Every element in the codomain is the image of at least one element in the domain. The range equals the codomain. |
| Bijective | A function that is both Injective and Surjective. | Each element in the domain maps to exactly one unique element in the codomain, and every element in the codomain is mapped to by exactly one element in the domain. |
| Constant Function | $\forall x f (x) = k$ (where $k$ is a fixed value) | The function always outputs the same value for every input. |
Additional Information: Understanding Quantifiers
The definitions use quantifiers which are important in mathematical logic:
- $\forall$ (For all / For every): This is the universal quantifier. It means the statement holds true for every element in the specified set (e.g., the domain or codomain).
- $\exist$ (There exists): This is the existential quantifier. It means there is at least one element in the specified set for which the statement holds true.
Understanding these quantifiers is key to interpreting formal definitions of function properties.
For example, $\forall x \forall y$ means "for any choice of x and any choice of y". $\forall y \exist x$ means "for any given y, we can find at least one x".
Important Questions from Relations - Teaching
Let ∈ = 0.0005, and Let Re be the relation {(x, y) = R 2∶ |x − y| < ∈}, Re could be interpreted as the relation approximately equal . Re is
(A) Reflexive
(B) Symmetric
(C) transitive
Choose the correct answer from the options given below:
Consider the following properties with respect to a flow network G = (V, E) in which a flow is a real-valued function f :
V × V → R
P 1: For all u,v ε V, f(u, v) = -f(v, u)
P 2: Σ v ε V f(u, v) = 0 for all u ε V
Which one of the following is/are correct?
Consider the following properties:
A. Reflexive
B. Antisymmetric
C. Symmetric
Let A = {a, b, c, d, e, f, g} and R= {(a, a),(b, b),(c, d),(c, g),(d, g),(e, e),(f, f),(g, g)} be a relation on A. Which of the following property (properties) is (are) satisfied by the relation R ?
