MAT $221$

BINARY OPERATIONS WORKSHEET

Note: This worksheet inchedes some of questions

from Tutorial Sheet $6($Binary Operations$).$

For the following examples answer the questions given at

the end.

b on $R,a**b=\frac{a+b}{2}$

$2$ on $Z^{+},a**b=a-b$

$3$ on $Z^{+},a**b=a^b$

$4$ on $R,a**b=a-b$

$5$ on $Z^{+},a**b=c,$ where $c$ is the largest

integer less than $ab.$

$($for eg$.2**7=13).$

$6$ on $N,a**b=gcd(a,b)$

$M_{23}(Q)-$ all $23$ matrices with entries from $Q.$

on $M_{2**3}(Q),A**B=A+B,A,Bin{M}_{23}(Q)$

$8$ on $\frac{R}{\frac{?}{?}}\{1\},a**b=a+b-ab$

$q$ on $\frac{R}{\frac{?}{?}}\{-1\},a**b=a+b+ab$

$10$ Let $Ql$ be the set of all finite: subsets of $Z.$

$)=(\{x_1,$dots,$x_n\},x_i$inZ

Define an operation on $Ul$ as follows: Let $A,$BinQl.

$A**B=(A-B)(B-A)$

Draw a Venn diagram.

Is $*$ a binary operati$$n$?$

Prove that $(A-B)(B-A)=(AB)-(AB).$

For example

$\{-2,0,1,3,4,7\}**\{-9,-7,4,7,8\}$

$=\{-2,0,1,3\}\{-9,-7,8\}$

$=\{-2,0,1,3,-9,-7,8\}.$