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Terms in this set (59)
Cartesian Product of A and B
denoted AxB and read "A cross B" is the set of all ordered pairs (a,b) where a is in A and b is in b symbolically.
AxB = { (a,b) | a [- A and b [- B }
Ordered Pair
(a,b) = (c,d) means that a=c and b=d
Proper Subset
A is a proper subset of B if, and only if, every element of A is in B, but there is at least one element of B that is not in A.
Universal Statement
Says that a certain property is true for all elements in a set.
Example: All positive numbers are greater than zero.
{x [- S | P(x)}
the set of all elements x in S such that P(x) is true.
Set Builder Notation x [- S
x is an element of S
x (not) [- S
x is not an element of S
Existential Statement
Given a property that may or may not be true, there is at least one thing for which the property is true.
Example: There is a prime number that is even.
Conditional Statement
Says that if one thing is true, then some other thing also has to be true.
Example: If 378 is divisible by 18, then it is also divisible by 6.
Subset A _C_ B
For all elements in x, if x [- A, then x [- B.
Nonsubset A (not) _C_ B
There is at least one element x, such that x [- and x (not) [- B.
Relation R from A to B
subset of AxB. Given an ordered pair (x,y) in AxB, x is related to y by R, written x R y, if, and only if, (x,y) is in R. The set A is called the domain of R and the set B is called the co-domain.
Function F from a set A to a set B
a relation with domain A and co-domain B that satisfies the following two properties:
1. For every element x in A, there is an element y in B such that (x,y) [- F.
2. For all elements x in A and y and z in B, if (x,y) [- F and (x,z) [- F, then y = z.
F(x)
If A and B are sets and F is a function from A to B, then given any element x in A, the unique element in B that is related to x by F is denoted as F(x), which is read "F of x"
Statement
A sentence that is true or false, but cannot be both.
~
denotes not.
^
denotes and.
~^
denotes or.
negation for p.
~p. Not p.
conjunction of p and q.
p^q.
disjunction
p (down arrow) q.
statement form
is an expression made up of statement variables (such as p, q, and r) and logical connectives (such as ^, ~, and down arrow) that becomes a statement when actual statements are substituted for the component statement variables.
Truth Table
For a given statement form displays the true values that correspond to all possible combinations of truth values for its component statement variables.
Logically Equivalent
Two statements are logically equivalent if, and only if, they have identical truth values for each possible substitutions of statements for their statement variables.
De Morgan's Laws
1. The negation of an and statement is logically equivalent to the or statement in which each component is negated.
2. The negation of an or statement is logically equivalent to the and statement in which each component is negated.
Tautology
A statement form that is always true regardless of the truth values of the individual statements substituted for its statement variables.
Tautological Statement
A statement whose form is a tautology.
Contradiction
A statement from that is false regardless of the truth values of the individual statements substituted for its statement variables.
Contradictory Statement
A statement whose form is a contradiction.
Conditional of p and q
If p, then q.
Contrapositive
The contrapositive of a conditional statement of the form "if p, then q," is "if ~q then ~p."
Symbolically: The contrapositive of p --> q is ~q --> ~p.
Conditional & Contrapositive
A conditional statement is logically equivalent to its contrapositive.
Converse
Suppose a conditional statement of the form "If p then q" is given.
The converse is "if q then p."
Inverse
Suppose a conditional statement of the form "If p then q" is given.
The inverse is "if ~p then ~q."
Only if statement
If p and q are statements, p only if q means "if not q then not p." or "if p then q"
Biconditional Statements
"p if, and only if, q" is denoted as p <-->. It is true if both p and q have the same truth values and is false if p and q have opposite truth values.
Sufficient conditional
If r and s are statements, r is a sufficient condition for s means "if r then s."
Necessary conditional
If r and s are statements, r is a necessary condition for s means "if not r then not s."
Sound
An argument that is called sound if, and only if, it is valid and all its premises are true.
Unsound
An argument that is not sound.
Contradiction rule
If you can show that the supposition that statement p is false leads logically to a contradiction, then you can conclude that p is true.
Predicate
A sentence that contains a finite number of variables and becomes a statement when specific values are substituted for the variables.
Domain
The domain of the predicate variable is the set if all values that may be substituted in place of the variable.
Truth Set
If P(x) is a predicate and x has domain D, the truth set of P(x) is the set of all elements of D that make P(x) true when they are substituted for x. The truth set is denoted as {x [- D | P(x)}
Quantifiers
Words that refer to quantities such as "some" or "all" and tell for how many elements a given predicate is true.
Negation of a Universal Statement
~((upside down A)x [- D , Q(x)) = (Backwards E)x [- D, such that ~Q(x)
Negation of an Existential Statement
~~((Backwards E)x [- D, such that Q(x) = (upside down A)x [- D , ~Q(x))
Even
2k
Odd
2k+1
Prime
An integer n is prime if, and only if, n>1 and for all positive integers r and s, if n=rs, then either or s equals n.
Composite
An integer n is composite if, and only if, n>1 and n=rs for some integers r and s with 1<r<n and 1<s<n
Counterexample
To disprove a statement, find a value of x in D for which the hypothesis P(x) is true and the conclusion Q(x) is false. Such an x is called an counterexample.
Direct Proof
1. Express the statement to be proven
2. Start the proof by supposing x is a particular but arbitrary chosen element of D for which the hypothesis is true.
3. Show that the conclusion Q(x) is true by using definitions, previously established results, and the rules for logical inference.
Contradiction Proof
1. Suppose the statement to be proved is false. That is, suppose the negation of the statement.
2. Show that this supposition leads logically to a contradiction.
3. Conclude that the statement to be proved is true.
Contraposition Proof
1. Express the statement to be proven.
2. Rewrite the statement in the contrapositive form.
3. Prove the contrapositive by a direct proof.
Sequence
A sequence is a function whose domain is either all the integers between two given integers or all the integers greater than or equal to a given integer.
Product notation
If m and n are integers and m<= n, the symbol n with n on top and below k=m, and a sub k on the right., which reads product from k equals m to n of a-sub-k., is the product of all the terms.
n factorial
Product of all the integers from 1 to n.
Mathematical Induction
Let P(n) to be a property that is defined for integers n, and let a be a fixed integer. Suppose the following two statements are true:
1. P(a) is true.
2. For all integers k >= a if P(k) is true then P(k+1) is true.
Then the statement for all integers n>=a, P(n) is true.
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