Flashcards · 7 cards

Binary operations

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Rule
In $a * b$ , which number goes where in the rule?
Answer
The first number replaces $a$ , the second replaces $b$ . In a table, the row heading is the first number.
a * b = 2a + b
5 * 3 = 2(5) + 3 = 13
Following the rulea * b: the first number replaces a, the second replaces b
From the lesson: Binary operations
Know it
How do you show an operation is commutative?
Answer
Swap the letters in the rule and simplify: if $a * b = b * a$ for all $a$ and $b$ , it is commutative.
From the lesson: Binary operations
Rule
How do you show an operation is associative?
Answer
Work out $(a * b) * c$ and $a * (b * c)$ in general, with letters. One set of numbers giving equal answers proves nothing.
Two ways to bracketLeft: a * b first, then with c. Right: b * c first, then a with it. Associative means they always agree
From the lesson: Binary operations
Rule
What is the identity, and how do you find it?
Answer
$e$ with $a * e = e * a = a$ for every $a$ . From a rule, solve $a * e = a$ ; in a table, its row and 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
The identity in a tableIts row and column repeat the headings
From the lesson: Binary operations
Know it
What is the inverse of $a$ ?
Answer
The element $a^{-1}$ with $a * a^{-1} = a^{-1} * a = e$ . Find the identity first.
From the lesson: Binary operations
Know it
When is an operation closed on a set?
Answer
When every answer it gives is back in the set.
From the lesson: Binary operations
Which method?
WAEC 2022 · Paper 2 · Q1
A binary operation $*$ is defined on the set $T = \{-2, -1, 1, 2\}$ by $p * q = p^2 + 2pq - q^2$ , where $p, q \in T$ .
Copy and complete the table.

$*$ $-2$ $-1$ $1$ $2$

$-2$ $7$ $-8$

$-1$ $2$ $-2$

$1$ $-7$ $1$

$2$ $-1$

Using the table in (a), find the value of $p$ such that $(-2 * p) * 2 = -7$ .
How do you fill in each entry?
Answer
Put the row heading in for $p$ and the column heading in for $q$ in $p^2 + 2pq - q^2$ .
From the lesson: Binary operations 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

$*$	$-2$	$-1$	$1$	$2$
$-2$		$7$		$-8$
$-1$		$2$	$-2$
$1$	$-7$			$1$
$2$		$-1$