Relations

Relations

 

What is a Relation?

 

In mathematics, a relation is a connection defined between two sets according to a specific rule. If \( A \) and \( B \) are any two sets, a relation from set \( A \) to set \( B \) is defined as: \[R \subset A \times B \Rightarrow R\; (Relation) \]

For example, if the sets
\[
A = \{1, 2, 3\}, \quad B = \{4, 5, 6\}
\]
are given, a relation \( R\) can be as follows:
\[
R= \{(1,4), (2,5), (3,6)\}
\]

Let \( A \) and \( B \) be two non-empty sets,

Every subset of \( A \times B \) is called a relation from A to B,

Every subset of \( B \times A \) is called a relation from B to A,

Every subset of \( A \times A \) is called a relation from A to A (a relation on A).

 

 

Examples:

 

Let the sets \( A = \{ 1, 2, 3 \} \) and \( B = \{ a, b, c \} \) be given.

\( \bullet \) If \( R_1 = \{ (1, a) , (2,a), (3,b) \} \) then \( R_1 \subset A \times B \) holds. Therefore, \(R_1 \) is a relation from \(A \) to \(B \).

\( \bullet \) If \( R_2 = \{ (2, a) , (3,b), (1,c) , (a, 1)\} \) since \( (a, 1 ) \) \( \notin A \times B\) then \( R_2 \not\subset A \times B \) holds.

\( \bullet \) If \( R_3 = \{ (a, a) , (a,b), (b,c) \} \) then \( R_3 \subset B \times B \) holds. Therefore, \( R_3\), is a relation from \(B \) to \(B \).

 

Note:

 

Let the given relation be:

\[
R \subset A \times B
\]

If \((x, y) \in R\), this can be denoted as
\[
y \, R \, x
\]
and it is said that the element \( x \) is mapped to \( y \) under the relation \( R \) (the element \( y \) is related to \( x \) under the relation \( R \)).

 

Ordered Pair

 

An element obtained by writing any two elements, such as \(a \) and \( b \), in a specific order as \((a, b) \) is called an ordered pair or simply a pair. In the ordered pair \((a, b) \), \(a\) is called the first component and \(b \) is called the second component.

In an ordered pair, the order of the components is essential. If the order of the components changes, a different pair is obtained. Therefore, \( (a, b) ≠(b,a) \) holds. Furthermore,

\[
\begin{aligned}
&(x_1, x_2, x_3) \quad \text{ordered triple} \\
&(x_1, x_2, x_4)\quad \text{ordered quadruple}\\
&\cdots \cdots\\
&\cdots \cdots\\
&(x_1, x_2, x_4, \cdots x_n)\quad \text{is called an ordered n-tuple.}\\
\end{aligned}
\]

 

Equality of Ordered Pairs:

 

For ordered pairs to be equal, their corresponding components must be equal to each other.

\[ (a, b) = (x,y ) \Rightarrow a = x \quad \text{and } \;\; y= b \;\; \]

 

Question 1

 

If \[ (2^x, 2^{y+1} )= (8, 2^{x-1}) \] what is the value of \(y \)?

 

\[
\text{A)} 0 \quad
\text{B) } 1 \quad
\text{C) } 2 \quad
\text{D) } 3 \quad
\text{E) } 4
\]

 

Solution:

 

 

Since \[ (2^x, 2^{y+1} )= (8, 2^{x-1}) \]

\[ 2^x = 8 \Rightarrow 2^x = 2^3 \Rightarrow x= 3 \;\; \text{and} \]

\[ 2^{y+1} = 2^{x-1} \Rightarrow y+1= x -1 \]

\[ \Rightarrow y+1= 3 -1 \Rightarrow y= 1 \]

\(\textbf{Answer: B} \)

 

Question 2

 

If \[ (\frac{1}{x}, \frac{2}{y}, \frac{4}{z} )= (y^2, z^2, x^2) \] what is the product \( xyz \)?

 

\[
\text{A)} 2 \quad
\text{B) } 3 \quad
\text{C) } 4 \quad
\text{D) } 5 \quad
\text{E) } 6
\]

 

Solution:

 

For ordered triples to be equal, their corresponding components must be equal to each other. Accordingly,

\[ \frac{1}{x} = y^2 \Rightarrow 1 = xy^2 \]

\[ \frac{2}{y} = z^2 \Rightarrow 2 = yz^2 \]

\[ \frac{4}{z} = x^2 \Rightarrow 4 = zx^2 \]

\[
\begin{aligned}
&1 = xy^2\\
&2 = yz^2\\
\times \quad &4 = zx^2 \\
\hline
\quad &8= x^3y^3z^3 \Rightarrow xyz= 2
\end{aligned}
\]

 

\(\textbf{Answer: A} \)

 

 

Applications of Relations

 

Relations are widely used in mathematics in areas such as functions, graph theory, set theory, and databases.

  1. Mathematics → Equivalence relations, orderings, functions.
  2. Logic and Set Theory → Demonstrates relationships between elements.
  3. Databases (SQL) → Connections between tables are modeled using relations.
  4. Graph Theory → Defines connections between nodes.

 

 

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