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Found in: Page 582

### Discrete Mathematics and its Applications

Book edition 7th
Author(s) Kenneth H. Rosen
Pages 808 pages
ISBN 9780073383095

# Let $${R_1} = \{ (1,2),(2,3),(3,4)\}$$ and $${R_2} = \{ (1,1),(1,2),(2,1),(2,2),(2,3),$$$$(3,1),(3,2),(3,3),(3,4)\}$$ be relations from $$\{ 1,2,3\}$$ to $$\{ 1,2,3,4\}$$. Finda) $${R_1} \cup {R_2}$$.b) $${R_1} \cap {R_2}$$.c) $${R_1} - {R_2}$$.d) $${R_2} - {R_1}$$.

(a) $${R_1} \cup {R_2} = \{ (1,1),(1,2),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3),(3,4)\}$$

(b) $${R_1} \cap {R_2} = \{ (1,2),(2,3),(3,4)\}$$

(c) $${R_1} - {R_2} = \emptyset$$

(d) $${R_2} - {R_1} = \{ (1,1),(2,1),(2,2),(3,1),(3,2),(3,3)\}$$

See the step by step solution

## Step 1: Given Data

$$\begin{array}{l}{R_1} = \{ (1,2),(2,3),(3,4)\} \\{R_2} = \{ (1,1),(1,2),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3),(3,4)\} \end{array}$$

## Step 2: Concept of the union, intersection and difference

Union $$A \cup B$$: All elements that are either in $$A$$or in $$B$$

Intersection $$A \cap B$$: All elements that are both in $$A$$and in $$B$$.

Difference $$A - B$$: All elements in $$A$$ that are NOT in $$B$$

## Step 3: Determine the value of $${R_1} \cup {R_2}$$

(a)

The union of two relations contains all ordered pairs that are in either relation.

$${R_1} \cup {R_2} = \{ (1,1),(1,2),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3),(3,4)\}$$

## Step 4: Determine the value of $${R_1} \cap {R_2}$$

(b)

The intersection of two relations contains all ordered pairs that are in both relations. We note that all ordered pairs in $${R_1}$$also occur in $${R_2}$$.

$${R_1} \cap {R_2} = \{ (1,2),(2,3),(3,4)\}$$

## Step 5: Determine the value of $${R_1} - {R_2}$$

(c)

$${R_1} - {R_2}$$contains all ordered pairs that are in the relation $${R_1}$$ that do not occur in the relation $${R_2}$$. We note that all ordered pairs in $${R_1}$$ also occur in $${R_2}$$, thus the difference $${R_1} - {R_2}$$ does not contain any elements.

$${R_1} - {R_2} = \emptyset$$

## Step 6: Determine the value of $${R_2} - {R_1}$$

(d)

$${R_2} - {R_1}$$ contains all ordered pairs that are in the relation $${R_1}$$that do not occur in the relation $${R_2}$$.

$${R_2} - {R_1} = \{ (1,1),(2,1),(2,2),(3,1),(3,2),(3,3)\}$$

Therefore,

(a) $${R_1} \cup {R_2} = \{ (1,1),(1,2),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3),(3,4)\}$$

(b) $${R_1} \cap {R_2} = \{ (1,2),(2,3),(3,4)\}$$

(c) $${R_1} - {R_2} = \emptyset$$

(d) $${R_2} - {R_1} = \{ (1,1),(2,1),(2,2),(3,1),(3,2),(3,3)\}$$