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architkap
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If the operation * is defined for all integers a and b by a*b = a+b-ab 31 May 2026, 07:12
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If the operation * is defined for all integers a and b by a*b = a+b-ab, which of the following statements could be true?

I. a*b=b*a

II. a*0=a

III. (a*b)*c = a*(b*c)

A. I

B. II

C. I and II

D. II and III

E. I, II and III
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E
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If the operation * is defined for all integers a and b by a*b = a+b-ab 31 May 2026, 08:36
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To determine which of the statements are true, let's analyze each one using the definition of the operation a∗b=a+b−ab:
Statement I: a∗b=b∗a
Let's find the expression for both sides:
  • Left-hand side (LHS): a∗b=a+b−ab
  • Right-hand side (RHS): b∗a=b+a−ba
Since addition and multiplication are commutative for integers (a+b=b+a and ab=ba), the expression b+a−ba is equivalent to a+b−ab. Thus, a∗b=b∗a. Statement I is true.
Statement II: a∗0=a
Substitute b=0 into the definition:
a∗0=a+0−a(0)
a∗0=a+0−0=a
Thus, a∗0=a. Statement II is true.
Statement III: (a∗b)∗c=a∗(b∗c)
Let's expand both sides to check for associativity:
  • Left-hand side (LHS):
    (a∗b)∗c=(a+b−ab)∗c
    Applying the operation again, treating (a+b−ab) as the first term and c as the second term:
    =(a+b−ab)+c−(a+b−ab)c
    =a+b−ab+c−ac−bc+abc
    =a+b+c−ab−ac−bc+abc
  • Right-hand side (RHS):
    a∗(b∗c)=a∗(b+c−bc)
    Applying the operation again, treating a as the first term and (b+c−bc) as the second term:
    =a+(b+c−bc)−a(b+c−bc)
    =a+b+c−bc−ab−ac+abc
    =a+b+c−ab−ac−bc+abc
Since the LHS expression simplifies to the exact same polynomial as the RHS expression, the operation is associative. Thus, (a∗b)∗c=a∗(b∗c). Statement III is true.
Since all three statements (I, II, and III) are true, the correct choice is E.
Correct Answer: E. I, II and III

architkap
If the operation * is defined for all integers a and b by a*b = a+b-ab, which of the following statements could be true?

I. a*b=b*a

II. a*0=a

III. (a*b)*c = a*(b*c)

A. I

B. II

C. I and II

D. II and III

E. I, II and III
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