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 ?
B and not A
Understanding Relation Properties: Reflexive, Antisymmetric, and Symmetric
Let's analyze the given relation R on the set A to determine which of the properties – Reflexive, Antisymmetric, and Symmetric – it satisfies. The set A is given as \(A = \{a, b, c, d, e, f, g\}\) and the relation R is given as \(R = \{(a, a),(b, b),(c, d),(c, g),(d, g),(e, e),(f, f),(g, g)\}\).
Checking for Reflexivity
A relation R on a set A is called reflexive if for every element \(x \in A\), the ordered pair \((x, x)\) is in R.
In this case, the set A contains the elements {a, b, c, d, e, f, g}. For R to be reflexive, it must contain the pairs \((a, a), (b, b), (c, c), (d, d), (e, e), (f, f), (g, g)\).
Let's check the pairs in R:
- \((a, a)\) is in R.
- \((b, b)\) is in R.
- \((c, c)\) is not in R.
- \((d, d)\) is not in R.
- \((e, e)\) is in R.
- \((f, f)\) is in R.
- \((g, g)\) is in R.
Since \((c, c) \notin R\) and \((d, d) \notin R\), the relation R is not reflexive. It does not satisfy property A.
Checking for Symmetry
A relation R on a set A is called symmetric if for every ordered pair \((x, y)\) in R, the ordered pair \((y, x)\) is also in R. That is, if \((x, y) \in R\), then \((y, x) \in R\).
Let's examine the pairs in R:
- \((a, a)\) is in R. For this pair, \(x=a\) and \(y=a\). \((y, x)\) is \((a, a)\), which is in R. This holds.
- \((b, b)\) is in R. \((b, b)\) is in R. Holds.
- \((c, d)\) is in R. For this pair, \(x=c\) and \(y=d\). Is \((d, c)\) in R? No, \((d, c) \notin R\).
Since we found a pair \((c, d) \in R\) such that \((d, c) \notin R\), the condition for symmetry is not met. The relation R is not symmetric. It does not satisfy property C.
We can also check the other non-loop pairs:
- \((c, g)\) is in R. Is \((g, c)\) in R? No, \((g, c) \notin R\).
- \((d, g)\) is in R. Is \((g, d)\) in R? No, \((g, d) \notin R\).
These pairs further confirm that R is not symmetric.
Checking for Antisymmetry
A relation R on a set A is called antisymmetric if for every ordered pair \((x, y)\) in R and the ordered pair \((y, x)\) in R, it must be the case that \(x = y\). That is, if \((x, y) \in R\) and \((y, x) \in R\), then \(x = y\).
Let's look for pairs \((x, y)\) and \((y, x)\) that are both in R where \(x \neq y\). If we find such a pair, the relation is not antisymmetric. If no such pair exists, the relation is antisymmetric.
Let's examine the pairs in R:
- \((a, a)\) is in R. \((a, a)\) is in R. Here \(x=y=a\). The condition \(x=y\) is met. This pair does not violate antisymmetry.
- \((b, b)\) is in R. \((b, b)\) is in R. Here \(x=y=b\). Condition met.
- \((c, d)\) is in R. Is \((d, c)\) in R? No. The premise "if \((x, y) \in R\) and \((y, x) \in R\)" is not met for \((c, d)\) and \((d, c)\). So this doesn't violate antisymmetry.
- \((c, g)\) is in R. Is \((g, c)\) in R? No. Doesn't violate antisymmetry.
- \((d, g)\) is in R. Is \((g, d)\) in R? No. Doesn't violate antisymmetry.
- \((e, e)\) is in R. \((e, e)\) is in R. Here \(x=y=e\). Condition met.
- \((f, f)\) is in R. \((f, f)\) is in R. Here \(x=y=f\). Condition met.
- \((g, g)\) is in R. \((g, g)\) is in R. Here \(x=y=g\). Condition met.
We did not find any pair \((x, y)\) with \(x \neq y\) such that both \((x, y)\) and \((y, x)\) are in R. The only time both \((x, y)\) and \((y, x)\) are in R is when \(x=y\). Therefore, the relation R is antisymmetric. It satisfies property B.
Summary of Findings
Based on our analysis:
- Relation R is not Reflexive (Property A not satisfied).
- Relation R is Antisymmetric (Property B satisfied).
- Relation R is not Symmetric (Property C not satisfied).
The relation R satisfies Antisymmetric but not Reflexive.
Comparing this with the given options:
- Only A (Reflexive) - Incorrect
- Only C (Symmetric) - Incorrect
- Both A (Reflexive) and B (Antisymmetric) - Incorrect
- B (Antisymmetric) and not A (Reflexive) - Correct
The property (properties) satisfied by the relation R is B and not A.
| Property | Condition | Satisfied by R? | Reason |
|---|---|---|---|
| Reflexive (A) | For all \(x \in A\), \((x, x) \in R\) | No | \((c, c) \notin R\), \((d, d) \notin R\) |
| Antisymmetric (B) | For all \((x, y) \in R\), if \((y, x) \in R\), then \(x=y\) | Yes | No pairs \((x, y)\) with \(x \neq y\) exist such that both \((x, y) \in R\) and \((y, x) \in R\) |
| Symmetric (C) | For all \((x, y) \in R\), \((y, x) \in R\) | No | \((c, d) \in R\) but \((d, c) \notin R\) |
Revision Table: Relation Properties Quick Check
| Property | How to Check | Example (from R or general) |
|---|---|---|
| Reflexive | Does every element in the set have a loop \((x, x)\) in the relation? | \(A = \{a, b\}\), \(R = \{(a, a), (b, b)\}\) is reflexive. \(R = \{(a, a)\}\) is not reflexive on A. |
| Symmetric | For every arrow from x to y, is there a corresponding arrow from y to x? If \((x, y)\) is a pair, is \((y, x)\) also a pair? | \(A=\{1,2\}\), \(R=\{(1,2),(2,1)\}\) is symmetric. \(R=\{(1,2)\}\) is not symmetric. |
| Antisymmetric | Are there any pairs \((x, y)\) and \((y, x)\) where \(x \neq y\)? If the only time both \((x, y)\) and \((y, x)\) appear is when \(x=y\), it's antisymmetric. | \(A=\{1,2\}\), \(R=\{(1,2)\}\) is antisymmetric. \(R=\{(1,2),(2,1)\}\) is not antisymmetric (because \(1 \neq 2\)). \(R=\{(1,1)\}\) is antisymmetric. |
Additional Information: Types of Relations
Understanding relation properties is fundamental in discrete mathematics and computer science. Relations can model various connections between elements of sets. Beyond the three properties discussed, relations can also be Transitive. A relation that is reflexive, symmetric, and transitive is called an Equivalence Relation. A relation that is reflexive, antisymmetric, and transitive is called a Partial Order Relation.
Visualizing relations on a finite set using directed graphs (digraphs) can help. Elements of the set are vertices, and ordered pairs in the relation are directed edges. Reflexivity means every vertex has a loop. Symmetry means if there is an edge from x to y, there is always an edge from y to x (edges come in pairs between distinct vertices). Antisymmetry means that between any two distinct vertices x and y, there can be at most one edge (either x to y or y to x, but not both).
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?
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
