Explanation From given table, it is observed that the set is { ! , ? , Δ , p } under binary operation ‘□’. Closure: There exist ? , ! ∈ { ! , ? , Δ , p } , Then ? ∥ ! is Δ which is an element belongs to the same set. That is, the system does not contain a symbol other than ! , ? , Δ , p . For any two elements from the system satisfies the closure property under an operation ‘□’. This shows the system satisfy closure property under an operation ‘□’. Thus, the system is closed under an operation‘□’. Identity: To obtain the identity element, check the identical rows and identical columns. The row of elements to the right of Δ is identical to the top row and the column of elements under Δ is identical to the far-left column. Therefore, Δ is the identity element of the system under an operation ‘□’. Inverse: If a element operates with its inverse, it will gives the result as the identity element under an operation ‘□’. From table it is observed that, the identity element is Δ . Since ?□!= Δ , ? is inverse of !. Since !□?= Δ , ! is inverse of ?. Since Δ □ Δ = Δ , Δ is its own inverse

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Chapter 9, Problem 16RE

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The property which fails to satisfy the system does not hold commutative group.

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Ch. 9.1 - Prob. 1E Ch. 9.1 - Prob. 2E Ch. 9.1 - Prob. 3E Ch. 9.1 - Prob. 4E Ch. 9.1 - Prob. 5E Ch. 9.1 - Prob. 6E Ch. 9.1 - Prob. 7E Ch. 9.1 - Prob. 8E Ch. 9.1 - Prob. 9E Ch. 9.1 - Prob. 10E

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The property which fails to satisfy the system does not hold commutative group.

Solution Summary: The author explains that the associative property fails to satisfy the system does not hold commutative group.

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