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proof exam 1

Terms in this set (38)

If-Then template
-Assume the hypothesis
-Write the conclusion at bottom, stating end result
-Unravel definitions in both directions
-Find what you know and what you need. Forge a link between the two halves.
If and Only If template (iff)
- (--->) Prove "If A, then B"
- (<---) Prove "If B, then A"
Counterexample Template
"If A, then B"
-Find an example where A is true, but B is false.
Inequality Template
-Start with a fact that is true
-Through algebraic manipulations, reach the desired conclusions
Logical Equivalence Template
-Construct a truth table showing the values of the two expressions for all possible values of the variables
-Check to see that two Boolean expressions always have the same value.
Proving two sets are equal template
-Suppose x ∈ A... therefore, x ∈ B
-Suppose x ∈ B.... therefore, x ∈ A
Therefore, A = B
Proving one set is a subset of another template
Let x ∈ A... therefore x ∈ B.
Therefore A ⊆ B.
Definition of Divisible
Let a and b be integers. If there exists an integer c such that b=ac, then a|b.
Definition of even
An integer is called even provided it is divisible by 2.
Definition of odd
An integer a where there exists an integer n, such that a = 2n+1.
Definition of prime
An integer P, provided P>1 and its only positive divisors are 1 and P.
Definition of Composite
An integer a, where there is an integer b such that 1<b<a and b|a.
Vacuous truth
If A is always false then the If-Then statement is true
If-Then
If A then B is synonymous with A implies B or B is implied by A. A-->B
If and Only If (IFF)
The statement A if and only if B is synonymous with A if B and B if A. A <--> B.
Boolean Variables
Variables that can only assume true and false as its values.
And : A ∧ B
Or : A ∨ B
Not : ¬A
Logical Equivalence
Two boolean expressions are equivalent if they have the same value on the same in/out.
Commutative Property (Boolean)
- x ∧ y = y ∧ x
- x ∨ y = y ∨ x
Associative Property (Boolean)
- (x ∧ y) ∧ z = x ∧ (y ∧ z)
- (x ∨ y) ∨ z = x ∨ (y ∨ z)
Zero Property (Boolean)
- x ∧ (¬x) = FALSE
- x ∨ (¬x) = TRUE
Identity Property (Boolean)
- x ∧ TRUE = x
- x ∨ FALSE = x
Nipotent Property (Boolean)
- x ∧ x = x
- x ∨ x = x
Distributive Property (Boolean)
- x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z)
- x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z)
DeMorgan's Law
- ¬(x ∧ y) = (¬x) ∨ (¬y)
- ¬(x ∨ y) = (¬x) ∧ (¬y)
Sets
An unordered collection of objects without (or ignoring) repeats
Elements
Let A be a set. The objects in set A are called the elements of A.
The empty Set
The set with no elements, denoted by ∅.
Subset
Let A and B be sets. When all elements of B are in A, denoted as B ⊆ A.
Definition of Rational Numbers
The rational numbers is the set of fractions and this set is written as Q. We can write this
set as {a/b: a ∈ Z, b ∈ Z, b != 0}
Power Set
Let A be a set. The power set of A is the set of all subsets of A. Written as 2^A.
Union
Let A and B be sets. The union of A and B is the set of all elements that are in A or
B. The union of A and B is denoted A ∪ B.
Intersection
Let A and B be sets. The intersection of A and B is the set of all elements that
are in both A and B. The union of A and B is denoted A ∩ B.
Commutative Property (sets)
- A ∪ B = B ∪ A
- A ∩ B = B ∩ A
Associative Property (Sets)
- A ∪ (B ∪ C) = (A ∪ B) ∪ C
- A ∩ (B ∩ C) = (A ∩ B) ∩ C
Distributive Property (Sets)
-A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) -A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
Zero Property (Sets)
- A ∪ ∅ = A
- A ∩ ∅ = ∅
Set Difference
Let A and B be sets. The set difference, A\B = {x : x ∈ A and x !∈ B}.
Symmetric Difference
The symmetric difference of A and B ,denoted A (triangle) B, is the set of all elements in A but not B or in B but not A. That is,
A (triangle) B = (A \ B) ∪ (B \ A)
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