C1-6 Flashcards
What is a set?
A collection of objects where there is a well defined way of determining whether a given object is included or excluded
What is the associative law of addition
placement of brackets in sum is irrelevant
a+(b+c) = (a+b)+c
What is the existence of additive identity
adding zero leaves number unchanged
a+0 = 0+a = a
What is the commutative law for addition
the order numbers are added doesn’t matter
a+b = b+a
What is the associative law of multiplication
the placement of brackets is irrelevant
a.(b.c) = (a.b).c
What is the commutative law of multiplication
the order numbers multiply doesn’t matter
a.b = b.a
What is the trichotomy law
any real number is either positive, negative or zero
What is closure under addition
If a and b are positive then a+b is positive
What is closure under multiplication
If a and b are positive then a.b is positive
What is this symbol and what does it mean?
∩
Intersection symbol
so A ∩ B contain all elements that are in both sets A and B (smaller list)
What is this symbol and what does it mean?
U
Union symbol
so A U B consists of elements that are in set A and set B (bigger list)
What is the existence of multiplicative identity
1 is the multiplicative identity- leaving the number unchanged
a.1 = 1.a = a
What is the existence of additive inverses
A number and its additive inverse sum to zero
a + (-a) = 0
What is the existence of multiplicative inverses
A number and its multiplicative inverse multiply to give 1
a . a^(-1) = 1
What is the distributive law
here the brackets play a crucial role
a. (b+c) = a.b + a.c
What are the natural numbers?
N - counting numbers
1, 2, 3, 4, …
What are integers?
Z- natural numbers, with their negatives and 0
0, 1, -1, 2, -2, …
What are the rational numbers?
a/b : a, b are integers, b does not equal 0
What is a hypothesis/ premise
something assumed or known to be true
What is a conclusion
outcome reached
What is the modulus of a: |a|
a, a greater than or equal to 0
-a, a is less than 0
also square root of the square of a.
What is a conjecture?
possible/ likely to be true
What is a theorem
conjecture proven to be true
What is a lemma
part of a proof of a theorem
what is a corollary
follows easily from a theorem
what is a proof
a chain of logical arguments between hypothesis and conclusion
What is V
read as “or”
either one or the other or both of the statements are true
What is ∧
read as “and”
both statements must be true
how to write: if p is true then q is true in notation
p => q
what is the converse of p => q
p <= q
or q => p
what is the contrapositive of p => q
¬q => ¬p
law of contraposition states implication and its contrapositive are equivalent when p => q is true then contrapositive is true.
what is the inverse of p => q
¬p => ¬q
what is the natural domain
the largest set of numbers for which a function f can be defined
when is a function injective
iff no two distinct points in Dom(f) have same image under f
injection never maps two elements
when is a function surjective
every element of B can be obtained by applying f to at least one element of A
when is a function bijective
both injective and surjective
bijection establishes a one-to-one correspondence between A and B
rules for modulus (for all x is real)
|x| = |-x|
x is less than or equal to |x|
-x is less than or equal to |x|
rules for modulus (or all x, y are real)
|xy| = |x| |y|
|x/y| = |x|/|y|
What is the triangle inequality
|x+y| is less than or equal to |x| + |y|
for all x,y ER
when is a function even
if f(-x) = f(x)
when is a function odd
f(-x) = -f(x)
(f+g) x
f(x) + g(x)
(f-g) x
f(x) - g(x)
(fg)x
f(x)g(x)
(f/g)x
f(x)/ g(x)
when is a function f: R–> R linear
if f(x+y)= f(x) = f(y) and f(cx) = cf(x)
sin (x+2π)
sin(x)
cos(x+2π)
cos(x)
what kind of function is sine
sine is odd
sin(-x) = =sin(x)
what kind of function is cosine
cosine is even
cos(-x) = cos(x)
sin (π/2 -x)
cos(x)
cos (π/2 -x)
sin(x)
sin (π-x)
sin(x)
cos(π-x)
-cos(x)
sin^2x + cos^2x
1
sin(2x)
2sin(x) cos(x)
cos (2x)
cos^2x - sin^2x
2cos^2x -1
1-2sin^2x
sin(x plus/ minus y)
sin(x) cos(y) plus/ minus cos(x) sin(y)
cos(x plus/ minus y)
cos(x) cos(y) minus/plus sin(x) sin(y)
tan(x)
sin(x)/ cos(x)
secx
cotx
cscx
1/ cosx
1/tanx
1/sinx
tan^2x + 1
sec^2x
1 + cot^2x
csc^2x
