A binary operation * on the set of real numbers is defined by a*b= a+b-ab for a,b, c is belongs to R.
i) show that the operation is associative. That is, (a*b)*c=a*(b*c) for all a,b,c belongs to R.
ii) Find the value of -5*8 and also the value of (4*6)*5/13.
your def: a*b= a+b-ab
= first + last - product of first times last
(a*b)*c
= (a+b-ab)*c
= a+b-ab + c - ac - bc + abc
= a+b+c - ac - ab - bc + abc
a*(b*c)
= a*(b+c - bc)
= a+b+c-bc - ab - ac + abc
= a+b+c - ac - ab - bc + abc
= (a*b)*c
so the operation is associative
-5*8 = -5+8 -(-40) = 43
I think you should try the last part yourself
remember, first + last - product
To show that the operation * is associative, we need to prove that for any real numbers a, b, and c, the equation (a * b) * c = a * (b * c) holds true.
(i) Proof of associativity:
Let's consider the left-hand side of the equation: (a * b) * c
Substituting the definition of * operation, we get:
(a * b) * c = (a + b - ab) * c
Expanding the brackets, we have:
(a + b - ab) * c = ac + bc - abc
Next, let's consider the right-hand side of the equation: a * (b * c)
Substituting the definition of * operation:
a * (b * c) = a * (b + c - bc)
Expanding the brackets, we obtain:
a * (b + c - bc) = a(b + c - bc) = ab + ac - abc
Comparing the expressions obtained for the left-hand side and the right-hand side, we see that they are equal (ac + bc - abc = ab + ac - abc). Thus, we have shown that (a * b) * c = a * (b * c), proving the associativity of the * operation.
(ii) Evaluating specific values:
To find the value of -5 * 8, we substitute a = -5 and b = 8 into the definition of * operation:
-5 * 8 = -5 + 8 - (-5)(8) = -5 + 8 + 40 = 43
To find the value of (4 * 6) * (5/13), we substitute a = 4, b = 6, and c = 5/13 into the * operation definition:
(4 * 6) * (5/13) = (4 + 6 - 4*6) * (5/13)
Simplifying the expression inside the brackets:
(10 - 24) * (5/13) = (-14) * (5/13) = -14 + 5 - (-14)(5/13)
Performing further calculations:
-14 + 5 - (-14)(5/13) = -14 + 5 + (70/13)
Adding fractions:
-14 + 5 + (70/13) = -9 + (70/13)
Finding a common denominator:
-9 + (70/13) = (-9 * 13 + 70) / 13
Simplifying the numerator:
(-117 + 70) / 13 = -47 / 13
Therefore, the value of (4 * 6) * (5/13) is -47/13.