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Q30E

Expert-verifiedFound in: Page 582

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 \(\{ 1,2,3,4\} \). Find**

**a) \({R_1} \cup {R_2}\).**

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

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

**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 \(B\)**

(a)

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

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

(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 \({R_2}\).

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

(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.

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

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

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