Show that addition, subtraction, and multiplication are binary operations on R, but division is not a binary operation on R. Further, show that division is a binary operation on the set R∗ of nonzero real numbers.
Solve to prove that addition, subtraction, and multiplication are binary operations on R, but division is not.
For Addition,
+:R×R→R
where (a,b)→a+b
For every real number a & b,
a+b is also a real number.
Hence, + is a binary operation on R.
For Subtraction
−:R×R→R
where (a,b)→a−b
For every real number a & b,a−b is also a real number.
Hence, − is a binary operation on R.
For Multiplication,
×:R×R→R
where (a,b)→a×b
For every real number a & b,
a×b is also a real number.
Hence, × is a binary operation on R .
For Division
÷:R×R→R
where (a,b)→a÷b
Here, a & b are real numbers
a÷b=ab
Let a=3 & b=0
ab=30 "Not defined"
Hence, ÷ is not a binary operation on R.
Solve to prove that division is a binary operation on the set R∗ of nonzero real numbers.
÷:R∗×R∗→R∗
" where (a,b)→a÷b
For every non-zero real number a & b
a÷b is also a non-zero real number.
For example:p=4 and q=−3
pq=−43∈R∗
Hence, ÷ is a binary operation on R∗