MAT112 Ch 7
7.1
Proposition – a statement that is wither true or false. Lowercase letters such as p, q, and r are used to
denote a proposition.
Notation
Meaning
not p
p or q
p and q
if p then q
Negation
Disjunction
T
F
– is false if p is true, and true if p is false.
F
T
T
T
F
F
– is true if one or both of p and q are true.
T
F
T
F
Conjunction
– is true if p and q are both true.
Conditional
– is false if p is true and q is false, but otherwise is true.
Converse Contrapositive -
T
T
F
F
T
F
T
F
T
T
T
F
T
F
F
F
T
T
F
F
T
F
T
F
T
F
T
T
is the converse of
is the contrapositive of
Tautology – each entry in its column in the truth table is T
Contradiction – each entry in its column in the truth table is F
Contingency – at least one entry is T and at least one entry is F
Logical Implication – For compound propositions P and Q, if whenever P is T, Q is also T, then P logically
implies Q. Denoted ⇒
Logical Equivalence – For compound propositions P and Q that have identical truth tables, we say that P
and Q are logically equivalent. Denoted
(Any conditional proposition is logically equivalent to its
contrapositive.)
MAT112 Ch 7
7.2
Element/Member of a Set – an object in a set. Denoted
(This states that x is an element of A)
Empty Set or Null Set – a set with no elements. Written as . (The Empty set is a subset of every set.)
Subset – Set A is a subset of set B if all of the elements of A are also elements of B. Sub set is written
as
. NOT a subset is written as
.
Equal Sets – Two sets have the exact same elements. Equal sets written as A = B. NOT equal sets written
as
(If
and
, then A = B.)
Union -
{ |
Intersection -
} Elements in the union only have to be in one or the other.
{ |
} Elements in the intersection must be in both sets.
Disjoint Sets – Sets S and B are disjoint if the sets have NO elements in common.
Universal Set – the set of all elements under consideration. Identified by the letter U.
Complement of a Set A - All of the elements in the universal set that are NOT in the set A. Written as A’.
{
}
|
7.3
Number of Elements in a Set – written as
Addition Principle (for Counting) – For any two sets A and B,
A and B are disjoint,
. If
Multiplication Principle (for Counting) – If two operations O1 and O2 are performed in order, with N1
and N2 possible outcomes respectively, then there are
possible combined outcomes of the first
operation followed by the second.
7.4
Permutation – a set of distinct objects arranged in a specific order without repetition.
Number of Permutations of n Objects – The number of permutations of n distinct objects without
repetition is
Permutations of n Objects taken r at a Time – An arrangement of r of the n objects in a specific order.
In this case BC and CB are two different objects and thus each one is counted.
Combination of n objects Taken r at a Time – an r-element subset of the set of n objects. The
arrangement of the elements in the subset does not matter. In this case BC is the same as CB and is not
counted both times.