Standard +0.8 This question requires students to construct a counterexample to disprove associativity, which demands understanding of what associativity means and creative problem-solving to find values where (x∆y)∆z ≠ x∆(y∆z). While only 2 marks, it's non-routine and requires genuine mathematical thinking rather than applying a standard procedure, placing it above average difficulty.
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]
\((3\Delta1)\Delta2 = -2\) and \(3\Delta(1\Delta2) = -4\)
M1
Searches for and finds correct counter-example to the associativity condition
\((3\Delta1)\Delta2 \neq 3\Delta(1\Delta2)\), therefore \(\Delta\) is not an associative binary operation, disproving Gary's claim
R1
Correctly argues that the binary operation is not associative
**Question 3:**
| Answer | Mark | Guidance |
|--------|------|----------|
| $(3\Delta1)\Delta2 = -2$ and $3\Delta(1\Delta2) = -4$ | M1 | Searches for and finds correct counter-example to the associativity condition |
| $(3\Delta1)\Delta2 \neq 3\Delta(1\Delta2)$, therefore $\Delta$ is not an associative binary operation, disproving Gary's claim | R1 | Correctly argues that the binary operation is not associative |
---
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]\\
\hfill \mbox{\textit{AQA Further AS Paper 2 Discrete Q3 [2]}}