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1st year MM101 > C1-6 > Flashcards

C1-6 Flashcards

1
Q

What is a set?

A

A collection of objects where there is a well defined way of determining whether a given object is included or excluded

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2
Q

What is the associative law of addition

A

placement of brackets in sum is irrelevant

a+(b+c) = (a+b)+c

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3
Q

What is the existence of additive identity

A

adding zero leaves number unchanged

a+0 = 0+a = a

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4
Q

What is the commutative law for addition

A

the order numbers are added doesn’t matter

a+b = b+a

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5
Q

What is the associative law of multiplication

A

the placement of brackets is irrelevant

a.(b.c) = (a.b).c

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6
Q

What is the commutative law of multiplication

A

the order numbers multiply doesn’t matter

a.b = b.a

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7
Q

What is the trichotomy law

A

any real number is either positive, negative or zero

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8
Q

What is closure under addition

A

If a and b are positive then a+b is positive

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9
Q

What is closure under multiplication

A

If a and b are positive then a.b is positive

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10
Q

What is this symbol and what does it mean?

∩

A

Intersection symbol

so A ∩ B contain all elements that are in both sets A and B (smaller list)

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11
Q

What is this symbol and what does it mean?

U

A

Union symbol

so A U B consists of elements that are in set A and set B (bigger list)

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12
Q

What is the existence of multiplicative identity

A

1 is the multiplicative identity- leaving the number unchanged

a.1 = 1.a = a

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13
Q

What is the existence of additive inverses

A

A number and its additive inverse sum to zero

a + (-a) = 0

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14
Q

What is the existence of multiplicative inverses

A

A number and its multiplicative inverse multiply to give 1

a . a^(-1) = 1

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15
Q

What is the distributive law

A

here the brackets play a crucial role

a. (b+c) = a.b + a.c

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16
Q

What are the natural numbers?

A

N - counting numbers

1, 2, 3, 4, …

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17
Q

What are integers?

A

Z- natural numbers, with their negatives and 0

0, 1, -1, 2, -2, …

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18
Q

What are the rational numbers?

A

a/b : a, b are integers, b does not equal 0

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19
Q

What is a hypothesis/ premise

A

something assumed or known to be true

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20
Q

What is a conclusion

A

outcome reached

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21
Q

What is the modulus of a: |a|

A

a, a greater than or equal to 0
-a, a is less than 0

also square root of the square of a.

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22
Q

What is a conjecture?

A

possible/ likely to be true

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23
Q

What is a theorem

A

conjecture proven to be true

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24
Q

What is a lemma

A

part of a proof of a theorem

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25
Q

what is a corollary

A

follows easily from a theorem

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26
Q

what is a proof

A

a chain of logical arguments between hypothesis and conclusion

27
Q

What is V

A

read as “or”

either one or the other or both of the statements are true

28
Q

What is ∧

A

read as “and”

both statements must be true

29
Q

how to write: if p is true then q is true in notation

A

p => q

30
Q

what is the converse of p => q

A

p <= q

or q => p

31
Q

what is the contrapositive of p => q

A

¬q => ¬p

law of contraposition states implication and its contrapositive are equivalent when p => q is true then contrapositive is true.

32
Q

what is the inverse of p => q

A

¬p => ¬q

33
Q

what is the natural domain

A

the largest set of numbers for which a function f can be defined

34
Q

when is a function injective

A

iff no two distinct points in Dom(f) have same image under f

injection never maps two elements

35
Q

when is a function surjective

A

every element of B can be obtained by applying f to at least one element of A

36
Q

when is a function bijective

A

both injective and surjective

bijection establishes a one-to-one correspondence between A and B

37
Q

rules for modulus (for all x is real)

A

|x| = |-x|

x is less than or equal to |x|

-x is less than or equal to |x|

38
Q

rules for modulus (or all x, y are real)

A

|xy| = |x| |y|

|x/y| = |x|/|y|

39
Q

What is the triangle inequality

A

|x+y| is less than or equal to |x| + |y|

for all x,y ER

40
Q

when is a function even

A

if f(-x) = f(x)

41
Q

when is a function odd

A

f(-x) = -f(x)

42
Q

(f+g) x

A

f(x) + g(x)

43
Q

(f-g) x

A

f(x) - g(x)

44
Q

(fg)x

A

f(x)g(x)

45
Q

(f/g)x

A

f(x)/ g(x)

46
Q

when is a function f: R–> R linear

A 
if f(x+y)= f(x) = f(y) 
and f(cx) = cf(x)
47
Q

sin (x+2π)

A

sin(x)

48
Q

cos(x+2π)

A

cos(x)

49
Q

what kind of function is sine

A

sine is odd

sin(-x) = =sin(x)

50
Q

what kind of function is cosine

A

cosine is even

cos(-x) = cos(x)

51
Q

sin (π/2 -x)

A

cos(x)

52
Q

cos (π/2 -x)

A

sin(x)

53
Q

sin (π-x)

A

sin(x)

54
Q

cos(π-x)

A

-cos(x)

55
Q

sin^2x + cos^2x

A

1

56
Q

sin(2x)

A

2sin(x) cos(x)

57
Q

cos (2x)

A

cos^2x - sin^2x
2cos^2x -1
1-2sin^2x

58
Q

sin(x plus/ minus y)

A

sin(x) cos(y) plus/ minus cos(x) sin(y)

59
Q

cos(x plus/ minus y)

A

cos(x) cos(y) minus/plus sin(x) sin(y)

60
Q

tan(x)

A

sin(x)/ cos(x)

61
Q

secx
cotx
cscx

A

1/ cosx
1/tanx
1/sinx

62
Q

tan^2x + 1

A

sec^2x

63
Q

1 + cot^2x

A

csc^2x

1st year MM101 (3 decks)

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