C1-6

Question 1

What is a set?

Answer 1

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

Question 2

What is the associative law of addition

Answer 2

placement of brackets in sum is irrelevant

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

Question 3

What is the existence of additive identity

Answer 3

adding zero leaves number unchanged

a+0 = 0+a = a

Question 4

What is the commutative law for addition

Answer 4

the order numbers are added doesn’t matter

a+b = b+a

Question 5

What is the associative law of multiplication

Answer 5

the placement of brackets is irrelevant

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

Question 6

What is the commutative law of multiplication

Answer 6

the order numbers multiply doesn’t matter

a.b = b.a

Question 7

What is the trichotomy law

Answer 7

any real number is either positive, negative or zero

Question 8

What is closure under addition

Answer 8

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

Question 9

What is closure under multiplication

Answer 9

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

Question 10

What is this symbol and what does it mean?

∩

Answer 10

Intersection symbol

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

Question 11

What is this symbol and what does it mean?

U

Answer 11

Union symbol

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

Question 12

What is the existence of multiplicative identity

Answer 12

1 is the multiplicative identity- leaving the number unchanged

a.1 = 1.a = a

Question 13

What is the existence of additive inverses

Answer 13

A number and its additive inverse sum to zero

a + (-a) = 0

Question 14

What is the existence of multiplicative inverses

Answer 14

A number and its multiplicative inverse multiply to give 1

a . a^(-1) = 1

Question 15

What is the distributive law

Answer 15

here the brackets play a crucial role

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

Question 16

What are the natural numbers?

Answer 16

N - counting numbers

1, 2, 3, 4, …

Question 17

What are integers?

Answer 17

Z- natural numbers, with their negatives and 0

0, 1, -1, 2, -2, …

Question 18

What are the rational numbers?

Answer 18

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

Question 19

What is a hypothesis/ premise

Answer 19

something assumed or known to be true

Question 20

What is a conclusion

Answer 20

outcome reached

Question 21

What is the modulus of a: |a|

Answer 21

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

also square root of the square of a.

Question 22

What is a conjecture?

Answer 22

possible/ likely to be true

Question 23

What is a theorem

Answer 23

conjecture proven to be true

Question 24

What is a lemma

Answer 24

part of a proof of a theorem

Question 25

what is a corollary

Answer 25

follows easily from a theorem

Question 26

what is a proof

Answer 26

a chain of logical arguments between hypothesis and conclusion

Question 27

What is V

Answer 27

read as "or" either one or the other or both of the statements are true

Question 28

What is ∧

Answer 28

read as "and" both statements must be true

Question 29

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

Answer 29

p => q

Question 30

what is the converse of p => q

Answer 30

p <= q | or q => p

Question 31

what is the contrapositive of p => q

Answer 31

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

Question 32

what is the inverse of p => q

Answer 32

¬p => ¬q

Question 33

what is the natural domain

Answer 33

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

Question 34

when is a function injective

Answer 34

iff no two distinct points in Dom(f) have same image under f | injection never maps two elements

Question 35

when is a function surjective

Answer 35

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

Question 36

when is a function bijective

Answer 36

both injective and surjective bijection establishes a one-to-one correspondence between A and B

Question 37

rules for modulus (for all x is real)

Answer 37

|x| = |-x| x is less than or equal to |x| -x is less than or equal to |x|

Question 38

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

Answer 38

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

Question 39

What is the triangle inequality

Answer 39

|x+y| is less than or equal to |x| + |y| | for all x,y ER

Question 40

when is a function even

Answer 40

if f(-x) = f(x)

Question 41

when is a function odd

Answer 41

f(-x) = -f(x)

Question 42

(f+g) x

Answer 42

f(x) + g(x)

Question 43

(f-g) x

Answer 43

f(x) - g(x)

Question 44

(fg)x

Answer 44

f(x)g(x)

Question 45

(f/g)x

Answer 45

f(x)/ g(x)

Question 46

when is a function f: R--> R linear

Answer 46

``` if f(x+y)= f(x) = f(y) and f(cx) = cf(x) ```

Question 47

sin (x+2π)

Answer 47

sin(x)

Question 48

cos(x+2π)

Answer 48

cos(x)

Question 49

what kind of function is sine

Answer 49

sine is odd sin(-x) = =sin(x)

Question 50

what kind of function is cosine

Answer 50

cosine is even cos(-x) = cos(x)

Question 51

sin (π/2 -x)

Answer 51

cos(x)

Question 52

cos (π/2 -x)

Answer 52

sin(x)

Question 53

sin (π-x)

Answer 53

sin(x)

Question 54

cos(π-x)

Answer 54

-cos(x)

Question 55

sin^2x + cos^2x

Answer 55

1

Question 56

sin(2x)

Answer 56

2sin(x) cos(x)

Question 57

cos (2x)

Answer 57

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

Question 58

sin(x plus/ minus y)

Answer 58

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

Question 59

cos(x plus/ minus y)

Answer 59

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

Question 60

tan(x)

Answer 60

sin(x)/ cos(x)

Question 61

secx cotx cscx

Answer 61

1/ cosx 1/tanx 1/sinx

Question 62

tan^2x + 1

Answer 62

sec^2x

Question 63

1 + cot^2x

Answer 63

csc^2x