Question -
Answer -
(i) Given On┬аZ+,defined * by┬аa┬а*┬аb┬а=┬аa┬атАУ┬аb
If a = 1 and b = 2 inZ+, then
a * b = a тАУ b
= 1 тАУ 2
= -1 тИЙ Z+┬а[because Z+┬аisthe set of non-negative integers]
For a = 1 and b = 2,
a * b тИЙ Z+
Thus, * is not abinary operation on Z+.
(ii) Given Z+,define * by┬аa*b┬а=┬аa b
Let a, b тИИ Z+
тЗТ a, b тИИ Z+
тЗТ a * b тИИ Z+
Thus, * is a binaryoperation on R.
(iii) Given on┬аR,define by┬аa*b┬а=┬аab2
Let a, b тИИ R
тЗТ a, b2┬атИИ R
тЗТ ab2┬атИИ R
тЗТ a * b тИИ R
Thus, * is a binaryoperation on R.
(iv) Given on┬аZ+┬аdefine* by┬аa┬а*┬аb┬а= |a┬атИТ┬аb|
Let a, b тИИ Z+
тЗТ | a тАУ b | тИИ Z+
тЗТ a * b тИИ Z+
Therefore,
a * b тИИ Z+, тИА a, b тИИ Z+
Thus, * is a binaryoperation on Z+.
(v) Given on Z+┬аdefine* by a * b = a
Let a, b тИИ Z+
тЗТ a тИИ Z+
тЗТ a * b тИИ Z+
Therefore, a * b тИИ Z+┬атИА a, b тИИ Z+
Thus, * is a binaryoperation on Z+.
(vi) Given On R,define * by a * b = a + 4b2
Let a, b тИИ R
тЗТ a, 4b2┬атИИ R
тЗТ a + 4b2┬атИИ R
тЗТ a * b тИИ R
Therefore, a *b тИИ R, тИА a, b тИИ R
Thus, * is a binaryoperation on R.