Binary Operations

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Step 1: Substitute a = 2 and b = 3. Step 2: Compute 2 * 3 = 2 + 3. Step 3: 2 + 3 = 5. Answer: 5 Check: The result is a real number, so the operation is closed on R.

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Step 1: Test closure on N. Step 2: Take a = 1 and b = 2. Step 3: 1 - 2 = -1. Step 4: -1 is not in N. Answer: No, it is not a binary operation on N.

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Step 1: Use the formula Number of binary operations on A = p^(p^2). Step 2: Here p = 3. Step 3: So the number is 3^(3^2) = 3^9. Answer: 3^9 Meaning: A binary operation is a function from A × A to A, s…

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Addition on R is a binary operation because any two real numbers give one unique real number. Example: 2 + 3 = 5. Square root is not a binary operation because it works on one number, not two. Example…

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Step 1: Find 2 * 3. 2 * 3 = 2(2) + 3 = 7. Step 2: Now find 7 * 4. 7 * 4 = 2(7) + 4 = 18. Answer: 18…

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Step 1: Compute 3 * 2. 3 * 2 = 3^2 = 9. Step 2: Compute 9 * 1. 9 * 1 = 9^1 = 9. Answer: 9 Important: The operation is exponentiation, so the order of evaluation matters.

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Use the closure law to test whether an operation is a binary operation on a set. Example: On Q, a * b = (ab)/2 is a binary operation because the result of any rational a and b is still rational. Quick…

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Step 1: Write a * b = a + b. Step 2: Write b * a = b + a. Step 3: Since a + b = b + a, both are equal. Answer: The operation is commutative. Also associative because (a + b) + c = a + (b + c).