Language of Mathematics, Sets, and Elementary Logic

Symbols in Mathematics:

1. Digits (0, 1, 2, …, 9)

2. Operations (+, -, x () [])

3. Variables - stand-in for the unknown (a, b, c, …, x, y, z)

4. Relation symbols (= ≠ < > ≤ ≥)

5. Special symbols (𝜋 θ ∑ ∞ ∀ ∃ ∅ U ∩ ∈ ⊆ ⊂ ℝ ⊥ ∠ ° % √)

Start of the alphabet (a, b, c,…)

Constants (with fixed values)

From i to n (i, j, k, l, m, n)

Positive integers (for counting)

End of the alphabet (…, x, y, z)

Variables (unknowns)

∑

∃

∞

∀

1. The sum of

2. There exists

3. infinity

4. for every (for any)

If the English language has Nouns (objects, fixed things), the language of Mathematics has?

Numbers

If the English language has Connectives/associations, the language of Mathematics has?

Operational symbols/grouping symbols

If the English language has verbs (to show comparison), the language of Mathematics has?

Relation symbols

If the English language has pronouns, the language of Mathematics has?

Variables (letters representing quantities)

It is an expression with a correct arrangement of mathematical symbols but does not express a complete thought.

Mathematical expression

It is one that expresses a complete thought. Moreover, it makes sense to ask if the sentence is TRUE or FALSE.

Mathematical sentence

plus, sum, total, increased by, more, more than, add, added

to, added by, added with, in addition to, combined with, put

together, augmented

+

minus, less, less than, difference, decreased by, diminished

by, subtracted from, subtracted by, exceeds by, lowered by,

reduced by, loss, fewer, exceeds

-

times, the product of, multiplied by, multiplied to, multiplied

with, twice, doubled, thrice, tripled, squared, cubed

.

ratio, quotient, divided by, half of, average, per, over, all over

÷

equals, is equal to, is the same as, is similar to, results in,

produces, represents, is equivalent to, is, are, was, were, will

be

=

It is a collection of distinct objects. The objects are called elements or members

Set

The objects must be _________, which means that we can tell whether any given object is or is not in the set.

well-defined

It means no duplication of the object in the set

Distinct

The symbol __ is used to denote that an object is an element of a set

∈

The set are enumerated and separated by comma.

Roster method or Tabulation Method

It is used to describe the elements of the set.

Rule Method or Set Builder Notation

A set whose elements are limited or countable

Example:

A = { 𝑥I𝑥 is a positive integer less than

A = {1,2,3, …,16}17}

Finite set

A set whose elements are unlimited or cannot be counted

Example:

D = { 𝑥I𝑥 is a negative odd integer}

D = { -1, -3, -5, . . .}

Infinite Set

A set with only one element

Example:

F = { 𝑥I𝑥 is an integer greater than 10 but less than 12}

F = {11}

Unit Set or Singleton Set

A set with no object or element

Example:

H = { 𝑥I𝑥 is an integer greater than -1 but less than 0}

H = { } or H = ∅

Empty Set or Null set

Two sets containing the same elements are equal.

Symbolically, D = E

Example:

D = { 𝑥I𝑥 is a natural number greater than 12 but less than 17}.

E = {15, 16, 14, 13} = {13, 14, 15, 16}

Equal Set

If Sets A and B have the same number of elements

Example:

A = {11,12, 13}

B = {x, y, z}

A=B

Equivalent Set

Two sets with no common elements

Example:

R = {2, 4, 6}

S = {11, 13, 15}

T = {2, 10, 25

It is the number of its elements, it is denoted by n(A) or IAl.

Cardinality number of a set

A is a subset of B if every element of A is in B. The phrases "A is contained in B" and "B contains A" are other ways of saying that A is a subset of B

Subset ⊆

If every element of A is in B, but there is at least one element in B that is not in A, then A is a ________ of B

Proper Subset ⊂

The number of proper subsets of a given set with "n" elements can be computed using the formula:

2^n - 1

All proper subsets are subsets, but not all subsets are…?

proper subsets

A set is always a subset of itself, but it is ____ a proper subset of itself.

never

The set of all subsets of a set A is called the _____ of A, denoted as P(A)

Power sets

It is the set containing all the possible elements under consideration or the set containing all the elements of its subsets, without repetition

Universal set

Sets A and B are __________ if they have no common elements, and their union is the universal set.

Complementary A’ or A^c

The _____ of two sets A and B is the set containing all the elements of both sets

Union ∪

The __________ of two sets A and B is the set containing the common elements of both sets

Intersection ∩

The ________ of two nonempty sets A and B is the set of all ordered pairs (a, b), where the first element a is from set A, and the second element b is from set B

Cross product x

It is the discipline that deals with the methods of reasoning. It is regarded as one of the oldest intellectual disciplines in the history of humanity

Logic

The Greek philosopher who pioneered logical reasoning

Aristotle

A proposition, sometimes called a…?

Declarative sentence that is either true or false, but not both.

It's Not a Question or Command

It Ends with a Period

Proposition

To represent propositions, the lower case letters like p,q,r,s,… are used. These are called…?

Propositional variables

It is a single statement that does not have other statements as parts.

Simple statement

It contains two or more statements joined by logical connectives.

Compound statement

Elementary Logic Table