8.03a Binary operations: and their properties on given sets

3 The function min \(( a , b )\) is defined by: $$\begin{aligned} \min ( a , b ) & = a , a < b \\ & = b , \text { otherwise } \end{aligned}$$ For example, \(\min ( 7,2 ) = 2\) and \(\min ( - 4,6 ) = - 4\). Gary claims that the binary operation \(\Delta\), which is defined as $$x \Delta y = \min ( x , y - 3 )$$ where \(x\) and \(y\) are real numbers, is associative as finding the smallest number is not affected by the order of operation. Disprove Gary's claim.
[0pt] [2 marks]

6 The group \(( G , \boldsymbol { A } )\) has the elements \(e , r , r ^ { 2 } , q , q r\) and \(q r ^ { 2 }\), where \(r ^ { 2 } = r \boldsymbol { \Delta } r , q r = q \boldsymbol { \Delta } r , q r ^ { 2 } = q \boldsymbol { \Delta } r ^ { 2 }\) and \(e\) is the identity element of \(G\). The elements \(q\) and \(r\) have the following properties: $$\begin{aligned} & r \boldsymbol { \Delta } r \boldsymbol { \Delta } r = e \\ & q \boldsymbol { \Delta } q = e \\ & r ^ { 2 } \boldsymbol { \Delta } q = q \boldsymbol { \Delta } r \end{aligned}$$ 6

1. 1. State the order of \(G\). 6
      1. (ii) Prove that the inverse of \(q r\) is \(q r\).
         6
   2. Complete the Cayley table for elements of \(G\). 6
   3. Complete the Cayley table for elements of \(G\).

      A	\(e\)	\(r\)	\(r ^ { 2 }\)	\(q\)	\(q r\)	\(q r ^ { 2 }\)
      \(e\)	\(e\)	\(r\)	\(r ^ { 2 }\)	\(q\)	\(q r\)	\(q r ^ { 2 }\)
      \(r\)	\(r\)	\(r ^ { 2 }\)	\(e\)
      \(r ^ { 2 }\)	\(r ^ { 2 }\)	\(e\)	\(r\)
      \(q\)	\(q\)	\(q r\)	\(q r ^ { 2 }\)	\(e\)
      \(q r\)	\(q r\)	\(q r ^ { 2 }\)	\(q\)	\(r ^ { 2 }\)
      \(q r ^ { 2 }\)	\(q r ^ { 2 }\)	\(q\)	\(q r\)	\(r\)	\(r ^ { 2 }\)	\(e\)

      6
   4. State the name of a group which is isomorphic to \(G\).

5

1. Describe the conditions necessary for a set of elements, \(S\), under a binary operation * to form a group.
   5
2. In the multiplicative group of integers modulo 13, the group \(G\) is defined as $$G = \left( \langle 10 \rangle , \times _ { 13 } \right)$$ 5 (b) (i) Explain why \(G\) is an abelian group.
   5 (b) (ii) Find the order of \(G\).
   5
3. State the identity element of \(G\) and prove it is an identity element. Fully justify your answer.
   5
4. Find all the proper non-trivial subgroups of \(G\), giving your answers in the form \(\left( \langle g \rangle , \times _ { 13 } \right)\), where \(g\) is an integer less than 13

9 The group \(\left( C , + _ { 4 } \right)\) contains the elements \(0,1,2\) and 3 9

1. 1. Show that \(C\) is a cyclic group.
      9
      1. (ii) State the group of symmetries of a regular polygon that is isomorphic to \(C\) 9
   2. The group ( \(V , \otimes\) ) contains the elements (1, 1), (1, -1), (-1, 1) and (-1, -1) The binary operation \(\otimes\) between elements of \(V\) is defined by $$( a , b ) \otimes ( c , d ) = ( a \times c , b \times d )$$ 9
   3. 1. Find the element in \(V\) that is the inverse of \(( - 1,1 )\) Fully justify your answer.
         [0pt] [2 marks]
         9
   4. (ii) Determine, with a reason, whether or not \(C \cong V\) \(\mathbf { 9 }\) (c) The group \(G\) has order 16
      Rachel claims that as \(1,2,4,8\) and 16 are the only factors of 16 then, by Lagrange's theorem, the group \(G\) will have exactly 5 distinct subgroups, including the trivial subgroup and \(G\) itself. Comment on the validity of Rachel's claim. \includegraphics[max width=\textwidth, alt={}, center]{5ff6e3bb-6392-49cf-b64d-23bc595cd92e-16_2493_1721_214_150}

6 The set \(S\) consists of the following four complex numbers. \(\begin{array} { l l l l } \sqrt { 3 } + \mathrm { i } & - \sqrt { 3 } - \mathrm { i } & 1 - \mathrm { i } \sqrt { 3 } & - 1 + \mathrm { i } \sqrt { 3 } \end{array}\) For \(z _ { 1 } , z _ { 2 } \in S\), the binary operation \(\bigcirc\) is defined by \(z _ { 1 } \bigcirc z _ { 2 } = \frac { 1 } { 4 } ( 1 + i \sqrt { 3 } ) z _ { 1 } z _ { 2 }\).

1. 1. Complete the Cayley table for \(( S , \bigcirc )\) given in the Printed Answer Booklet.
   2. Verify that ( \(S , \bigcirc\) ) is a group.
   3. State the order of each element of \(( S , \bigcirc )\).
2. Write down the only proper subgroup of ( \(S , \bigcirc\) ).
3. 1. Explain why ( \(S , \bigcirc\) ) is a cyclic group.
   2. List all possible generators of \(( S , \bigcirc )\).

6 The set \(S\) consists of all real numbers except 1. The binary operation * is defined for all \(a , b\) in \(S\) by $$a * b = a + b - a b$$

1. By considering the identity \(a + b - a b \equiv 1 - ( a - 1 ) ( b - 1 )\), or otherwise, show that \(S\) is closed under *.
2. Show that * is associative on \(S\).
3. Find the identity of \(S\) under \(*\), and the inverse of \(x\) for all \(x \in S\).
4. The set \(S\), together with the binary operation *, forms a group \(G\). Find a subgroup of \(G\) of order 2 .

The operation \(\circ\) on real numbers is defined by \(a \circ b = a|b|\).

1. Show that \(\circ\) is not commutative. [2]
2. Prove that \(\circ\) is associative. [4]
3. Determine whether the set of real numbers, under the operation \(\circ\), forms a group. [4]

The operation \(*\) is defined on the elements \((x, y)\), where \(x, y \in \mathbb{R}\), by $$(a, b) * (c, d) = (ac, ad + b).$$ It is given that the identity element is \((1, 0)\).

1. Prove that \(*\) is associative. [3]
2. Find all the elements which commute with \((1, 1)\). [3]
3. It is given that the particular element \((m, n)\) has an inverse denoted by \((p, q)\), where $$(m, n) * (p, q) = (p, q) * (m, n) = (1, 0).$$ Find \((p, q)\) in terms of \(m\) and \(n\). [2]
4. Find all self-inverse elements. [3]
5. Give a reason why the elements \((x, y)\), under the operation \(*\), do not form a group. [1]

The binary operation \(*\) is defined as $$a * b = ab + 1 \quad \text{where } a, b \in \mathbb{R}$$

1. Prove that \(*\) is commutative on \(\mathbb{R}\) [2 marks]
2. Prove that \(*\) is not associative on \(\mathbb{R}\) [3 marks]

The binary operation \(\nabla\) is defined as \(a \nabla b = a + b + ab\) where \(a, b \in \mathbb{R}\)

1. Determine if \(\nabla\) is commutative on \(\mathbb{R}\) Fully justify your answer. [2 marks]
2. Prove that \(\nabla\) is associative on \(\mathbb{R}\) [3 marks]

The binary operation \(\oplus\) acts on the positive integers \(x\) and \(y\) such that $$x \oplus y = x + y + 8 \pmod{k^2 - 16k + 74}$$ where \(k\) is a positive integer.

1. 1. Show that \(\oplus\) is commutative. [1 mark]
   2. Determine whether or not \(\oplus\) is associative. Fully justify your answer. [2 marks]
2. Find the values of \(k\) for which 3 is an identity element for the set of positive integers under \(\oplus\) [3 marks]