Let $R = \{(1, 3), (4, 2), (2, 4), (2, 3), (3, 1)\}$ be a relation on the set $A = \{1, 2, 3, 4\}$. The relation R is
Show Hint
Recall the types of relation
- a function
- transitive
- not symmetric
- reflexive
The Correct Option is C
Approach Solution - 1
Let \(R = \{(1, 3), (4, 2), (2, 4), (2, 3), (3, 1)\}\) be a relation on the set \(A = \{1, 2, 3, 4\}\).
Then
(a) Since \( (2, 4) \in R\) and \((2, 3) \in R\:\: \therefore R\) is not a function
(b) Since \((1, 3) \in R\) and \((3, 1) \in R\) but \((1, 1) \therefore R \:\: \)is not transitive.
(c) Since \((2, 3) \in R\) but \((3, 2) \notin\,R\,\therefore R\) is not symmetric.
(d) Since \((4, 4) \notin R \therefore R\) is not reflexive
\(\therefore\) \(R\) not symmetric holds.
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Approach Solution -2
The Correct Answer is (C)
Real Life Applications
The real-life examples of sets, relations, and functions are:
1. Sets: A set is a collection of objects. For example, the set of all odd numbers is a set. The set of all people in the world is also a set.
2. Relations: A relation is a set of ordered pairs. For example, the relation "is smaller than" is a relation on the set of all people. The relation "is friends with" is also a relation on the set of all people.
3. Functions: A function is a special type of relation in which each element of the domain is associated with exactly one element of the range. For example, the function "square" is a function on the set of real numbers. The function "cube" is also a function on the set of real numbers.
Question can also be asked as
1. What is the relation r?
2. What does the relation r represent?
3. How can the relation r be interpreted?
4. What are some real-life examples of the relation r?
Approach Solution -3
The Correct Answer is (C)
Sets are organized collections of objects or elements.
- Sets are composed of elements which could be letters, numbers, shapes, etc.
- A set is represented by a capital letter, for instance, A = {...}.
- Elements in a set are represented as A = {1, 2, 3, 4, 5}.
- In a set, all elements should be interrelated and share a common property.
Sets Formulas
Some important Sets of Formulas are listed as follows:
For any three sets P, Q, and R:
- n ( P ∪ Q ) = n(P) + n(Q) – n ( P ∩ Q)
- If P ∩ Q = ∅, then n ( P ∪ Q ) = n(P) + n(Q)
- n( P – Q) + n( P ∩ Q ) = n(P)
- n( Q – P) + n( P ∩ Q ) = n(Q)
- n( P – Q) + n ( P ∩ Q) + n( Q – P) = n ( P ∪ Q )
- n ( P ∪ Q ∪ R ) = n(P) + n(Q) + n(R) – n ( P ∩ Q) – n ( Q ∩ R) – n ( R ∩ P) + n ( P ∩ Q ∩ R)
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Concepts Used:
Relations and functions
A relation R from a non-empty set B is a subset of the cartesian product A × B. The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in A × B.
A relation f from a set A to a set B is said to be a function if every element of set A has one and only one image in set B. In other words, no two distinct elements of B have the same pre-image.
Representation of Relation and Function
Relations and functions can be represented in different forms such as arrow representation, algebraic form, set-builder form, graphically, roster form, and tabular form. Define a function f: A = {1, 2, 3} → B = {1, 4, 9} such that f(1) = 1, f(2) = 4, f(3) = 9. Now, represent this function in different forms.
- Set-builder form - {(x, y): f(x) = y2, x ∈ A, y ∈ B}
- Roster form - {(1, 1), (2, 4), (3, 9)}
- Arrow Representation
