Flashcards · 11 cards

Binary operations

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Rule
With $a * b = 3a - b$ , what are $4 * 5$ and $5 * 4$ ?
Answer
The first number replaces $a$ : $4 * 5 = 12 - 5 = 7$ , but $5 * 4 = 15 - 4 = 11$ .
a * b = 2a + b
5 * 3 = 2(5) + 3 = 13
Using the ruleThe first number replaces a, the second replaces b
From the lesson: Working out an operation
Rule
How do you work out $(a * b) * c$ ?
Answer
Brackets first: work out $a * b$ , then combine that answer with $c$ .
Brackets firstWork out a * b, then combine that answer with c
From the lesson: Working out an operation
Rule
When is an operation closed on a set?
Answer
When every answer is back in the set: every entry in its table is a member of the set.
⊗ 1 2 3 4
1 1 2 3 4
2 2 4 1 3
3 3 1 4 2
4 4 3 2 1
ClosedEvery entry is 1, 2, 3 or 4: ⊗ is closed on {1, 2, 3, 4}
From the lesson: Properties and tables
Rule
When is an operation commutative?
Answer
When $a * b = b * a$ for every $a$ and $b$ . In a table: symmetric about the leading diagonal.
⊗ 1 2 3 4
1 1 2 3 4
2 2 4 1 3
3 3 1 4 2
4 4 3 2 1
Commutative2 ⊗ 4 = 4 ⊗ 2 = 3: the table is symmetric about the leading diagonal
From the lesson: Properties and tables
Rule
When is an operation associative?
Answer
When $(a * b) * c = a * (b * c)$ for every $a$ , $b$ and $c$ .
AssociativeThe two ways of bracketing must always give the same answer
From the lesson: Properties and tables
Rule
What is the identity element, and how do you spot it in a table?
Answer
$e$ with $a * e = e * a = a$ for every $a$ . In a table, its row and its column repeat the headings.
⊗ 1 2 3 4
1 1 2 3 4
2 2 4 1 3
3 3 1 4 2
4 4 3 2 1
IdentityRow 1 and column 1 repeat the headings: 1 is the identity
From the lesson: Properties and tables
Know it
How do you find the identity of $a * b = a + b + 2ab$ ?
Answer
Solve $a * e = a$ : $a + e + 2ae = a$ , so $e(1 + 2a) = 0$ and $e = 0$ . Then check $e * a = a$ too.
From the lesson: Properties and tables
Rule
What is the inverse of an element $a$ ?
Answer
The element that combines with $a$ to give the identity: $a * a^{-1} = a^{-1} * a = e$ . Find the identity first.
⊗ 1 2 3 4
1 1 2 3 4
2 2 4 1 3
3 3 1 4 2
4 4 3 2 1
Inverse2 ⊗ 3 = 1, the identity, so the inverse of 2 is 3
From the lesson: Properties and tables
Rule
How do you solve $3 \otimes n = 2$ from a table?
Answer
Go along row 3 to the entry 2, then read its column heading: that is $n$ .
⊗ 1 2 3 4
1 1 2 3 4
2 2 4 1 3
3 3 1 4 2
4 4 3 2 1
Solving from a table3 ⊗ n = 2: along row 3, the 2 is in column 4, so n = 4
From the lesson: Properties and tables
Which method?
WAEC 2017 · Paper 2 · Q13 (a)
An operation $*$ is defined by $x * y = x + y + 2xy$ , $x, y \in \mathbb R$ . (i) Calculate $(2 * 3) * 5$ . (ii) Find the truth set of $(x * 7) = (x * 5) * 2$ .
Where do you start $(2 * 3) * 5$ ?
Answer
The bracket: $2 * 3 = 2 + 3 + 2(2)(3) = 17$ . Then work out $17 * 5$ with the same rule.
From the lesson: Working out an operation Try the question
Which method?
WAEC 2014 · Paper 2 · Q13 (b)
An operation $(*)$ is defined on the set $\mathbb R$ of real numbers by $m * n = \dfrac{-n}{m^2 + 1}$ , where $m, n \in \mathbb R$ . If $m = -3$ and $n = -10$ , show whether or not $(*)$ is commutative.
How do you show whether it is commutative?
Answer
Work out both $m * n$ and $n * m$ with the given values. If they differ, it is not commutative.
From the lesson: Properties and tables Try the question

⊗	1	2	3	4
1	1	2	3	4
2	2	4	1	3
3	3	1	4	2
4	4	3	2	1

⊗	1	2	3	4
1	1	2	3	4
2	2	4	1	3
3	3	1	4	2
4	4	3	2	1

⊗	1	2	3	4
1	1	2	3	4
2	2	4	1	3
3	3	1	4	2
4	4	3	2	1

⊗	1	2	3	4
1	1	2	3	4
2	2	4	1	3
3	3	1	4	2
4	4	3	2	1