algebra MAT1031

Question 1

cardinality meaning

Answer 1

measurement of elements within a set

Question 2

what does A \ B or A - B mean

Answer 2

objects that belong to A and not to B
e.g
A = {3,9,14},
B = {1,2,3},
A-B = {9,14}

Question 3

Z means..

Answer 3

integers e.g {…-4,-3,-2,-1,0,1,2,3,4…}

Question 4

N means…

Answer 4

natural numbers e.g {1,2,3,4…}

Question 5

N(o) means…

Answer 5

natural numbers including 0 e.g {0,1,2,3,4…}

Question 6

Q means…

Answer 6

rational numbers where {m/n|m∈Z, n∈N}

Question 7

R means…

Answer 7

real numbers (non-complex)

Question 8

C means

Answer 8

complex numbers

Question 9

what does ∈ mean

Answer 9

‘element of’ e.g A={3,9,14}, 3 ∈ A

Question 10

what does | mean

Answer 10

‘such that’ e.g {m/n|m∈Z, n∈N}

Question 11

describe the difference between ( and [

Answer 11

square brackets mean the end point is included, and round parentheses mean it’s excluded
e.g [4,9) 9 is not included

(dubble check)

Question 12

∀ means..
∃ means..

Answer 12

‘for all’
‘there exists’

Question 13

Ø means…

Answer 13

empty set

Question 14

A ⊆ B

A ⊊ B

Answer 14

A is a subset of B where A can = B

A is a sub set of B but A ≠ B (sometimes it also means A ≠ ∅)

(dubble check)

Question 15

: means…

Answer 15

‘is’ (may also be used in place of | where its meaning in ‘such that’)

Question 16

¬ means…

Answer 16

‘not’

Question 17

write irrational numbers w/ signs

Answer 17

R - Q (real numbers - ratinal numbers)

Question 18

what are the cardinalitys for these
A := ∅,
A := {∅},
A := {∅, {∅}}

state the element sets of A := {∅, {∅}}

Answer 18

0,1,2
0 as for something to be considered a set therefore count as a cardinality it must have brackests around it. (Dubble check this)

∅ and {∅} (an empty set and a set containing an empty set)

Question 19

p ⇒ q means…

what is the contrapositive of p ⇒ q

if p ⇒ q is true then, …

what is the basic process of Proof by contraposition

Answer 19

‘if p then q’

¬ ⇒ ¬p - ‘if not p the not q’

¬q ⇒ ¬p is true

the implication p ⇒ q is true if and only if its contraposition ¬q ⇒ ¬p is true. As such, proving ¬q ⇒ ¬p is equivalent to proving
p ⇒ q. There are cases when it is easier to prove the contrapositive rather than prove the statement itself directly

Question 20

what is assoiative

Answer 20

when more than two numbers are added or multiplied, the result remains the same, irrespective of how they are grouped. For instance, 2 × (7 × 6) = (2 × 7) × 6.

Question 21

what is a binary operatiokn

Answer 21

when the basic operations are preformed on to numbers e.g the addition and multiplication of natural numbers

(dubble check)

Question 22

what is commutative

Answer 22

A commutative (also binary operation) is an operation where a ∗ b = b ∗ a. Addition is a classic example: 3 + 4 = 4 + 3, since they both equal 7. although subtraction is not

Question 23

abalian group

Answer 23

a group G with a binary operation * is abelian if, for any a, b ∈ G, ab=ba

(dubble check)

Question 24

:= means…

Answer 24

“is equal by definition to”