MATM LESSON 2 Flashcards
a system of conventional spoken, manual (signed), or written symbols by means of which human beings, as members of a social group and participants initsculture, express themselves.
Language
Language itself is:
- Precise
- Concise
- Powerful
It can make very fine distinctions among set of symbols
Precise
It can briefly express long sentences
Concise
It gives upon expressing complex thoughts
Powerful
A ___ in mathematics is a collection of well defined and distinct objects, considered as an object in its own right. Sets are one of the most fundamental concepts in mathematics
set
- Provides a list of all elements
- Example {1,2,3,4}
- Suits for sets with a lesser number of elements
- Easy to understand
Roster Form
- Defines the set by using a logical condition
- Suits for sets with more elements
- Little tricky for non-math people
Set Builder Form
Tells how many things are in a set. When counting a set of objects, the last word in the counting sequence names the quantity for that set.
Cardinality
- the TOTALITY of ALL the elements in two or more given sets
- denoted by ‘‘U’’
Universal Set
has no elements, it is empty. Its cardinal number is zero.
Null Set
- is the combination of elements of two or more sets
- Example: A={1,2,3,4} and B={1,5,6,7}
then
A ∪ B = {1,2,3,4,5,6,7,}
UNION SET
- it refers to the common elements of the 2 given sets
-Example: A={a,b,c,d,e} and B={a,e,i,o,u}
then
A ∩ B = {a,e}
INTERSECTION OF SETS
- it refers to the elements of 1st set alone but not an element of other set.
- Example: Find A-B and B-A
A={1,2,3,4,5} and B={2,4,6,8,10}
then
A – B = {1,3,5}
B – A = {6,8,10}
DIFFERENCE OF SETS
- refers to the elements of the universal set alone which is not part of the concerned set.
- If U={a, b, c, d, e, f, g, h, u, j}
and A={a, e, i}
A’ ={b, c, d, f, g, h, j}
COMPLEMENTARY OF SETS
– a set is subset if all its elements are can be found to other sets.
- “⊆” – “is a subset of”. (equally the same)
“⊂” – “is a proper subset of”.
A={1,2,3,4} B ⊂ A
B={1,2,3} C ⊄ A
C={3,4,5} D ⊆ A
D={2,1,4,3}
SUBSET OF SETS
- sets are equal if they have the exactly the same elements
- M={1,3,9,5,−7} andN={5,−7,3,1,9} therefore M=N
Equal sets
- sets are equivalent if they have same number of elements
- ## S={1,2,3} andT={a,b,c} therefore S∼T
Equivalent Sets
serves as a set of rules that govern the structure and presentation of mathematical proofs.It allows us to determine the validity of arguments in and out of mathematics.
Logic
is a statement that is, by itself, either true or false. They can be expressed in symbols P, Q, R, or p, q, r.
proposition
- means single idea statement
a. Simple
- conveys two or more ideas
b. Compound
is a declarative sentence (that is, a sentence that declares a fact) that is either true or false, but not both.
Proposition
Not ~p
Negation
And P ^ Q
Conjunction
Or P V Q
Disjunction
If…Then P → Q
Conditional
If and only if P ↔ Q
Bi-conditional
“for all” or “for every”, denoted by ∀
Universal Quantifier
“there exists”, denoted by ∃
Existential Quantifier
p → q If p, then q
Conditional Statement
q → p If q, then p
Converse
~p → ~q If not p, then not q
Inverse
~q → p If not q, then not p
Contrapositive
If True then False
If False then True
Negation ~p
If theres two True, then True
If theres a single False then automatically False
Conjunction p ^ q
If theres two False, then False
If theres a single True then automatically True
Disjunction p v q
My Theory
If False is last then F
(T, F = T)
Conditional p → q
If different then False (T, F = F) (F,T = F)
If same then True ( F, F = T)
Bi conditional p ↔ q
