BINARY OPERATIONS

GALAXY EXPLORER

1 Binary Operation Check

Check if a*b = a·b on ℝ is a binary operation

✓ Binary Operation Verified

1. For any two real numbers a and b, their product a·b is always a real number

2. The operation is well-defined for all pairs in ℝ × ℝ

3. The result is always in the original set (ℝ)

Check if a*b = min{a,b} on A = {1,2,3,4,5} is binary

✓ Binary Operation Verified

1. For any two numbers in A, their minimum is always in A

2. The operation always gives a result in the same set

3. Example: 3*4 = min{3,4} = 3 ∈ A

Check if a*b = a√b on ℝ is binary

✗ Not a Binary Operation

1. When b is negative, √b is not a real number

2. The operation is not defined for all pairs in ℝ × ℝ

3. Counterexample: 2*(-1) = 2√(-1) ∉ ℝ

2 Power Operation

On ℤ, define ∗ by (m∗n) = mⁿ + nᵐ. Is ∗ binary on ℤ?

✗ Not a Binary Operation

1. For positive integers, it works (e.g., 2∗3 = 8+9 = 17 ∈ ℤ)

2. But for negatives:

  • (-1)∗(-2) = (-1)⁻² + (-2)⁻¹ = 1 + (-0.5) = 0.5 ∉ ℤ
  • 0⁰ is undefined

3. Some results fall outside ℤ or are undefined

3 Special Operation

On ℝ, define ∗ by (a∗b) = a + b + ab - 7. Is ∗ binary on ℝ? If so, find 3∗(-7/15)

✓ Binary Operation Verified

1. For any a, b ∈ ℝ, a + b + ab - 7 is always a real number

2. The operation is defined for all pairs in ℝ × ℝ

3. Calculation for 3∗(-7/15):

3 + (-7/15) + (3 × -7/15) - 7 = 3 - 0.4667 - 1.4 - 7 ≈ -5.8667