INTECH 1100 ( 2ND TERM)

An integer n is ____ if, and only if, n equals twice some integer. Symbolically, if n is an integer, then n is

even

what do you call this type of symbol?

∃ an integer k such that n = 2k.

even

An integer n is ____ if, and only if, n equals twice some integer plus 1

odd

what do you call this type of symbol?

odd ⇔ ∃ an integer k such that n = 2k + 1

odd

An integer n is ___ if, and only if, n > 1, and for all positive integers r and s, if n = rs, then either r or s equals n

prime

what do you call this type of symbol?

∀ positive integers r and s, if n = r s then either r = 1 and s = n or r = n and s = 1.

prime

An integer n is ___ if, and only if, n > 1 and n = rs for some integers r and s with 1 < r < n and 1 < s < n.

composite

what do you call this type of symbol?
∃ positive integers r and s such that n = r s and 1 < r < n and 1 < s < n.

composite

In proving existential statements are called___?

constructive proofs of existence

what do you call in Disproving Universal Statements

Counterexample

what do you call in Proving Universal Statements

proof by exhaustion

To show that every element of a set satisfies a certain property, suppose x is a particular but arbitrarily chosen element of the set, and show that x satisfies the property

Method of Generalizing from the Generic Particular

a step-by-step method for
performing some action.

ALGORITHM

used to refer to a specific storage location in a computer’s
Memory.

Variable -

indicates the set in which the the variable takes its values, whether the set of integers, real numbers, character strings, and so forth

Data type -

gives a value to a variable.

x:= e

Assignment statement -

allow this natural order to be overridden by using the current values of program variables to determine which algorithm statement will be executed next

Conditional statements -

Conditional statements are denoted either: ( give the form)

a. if (condition)
then s1
else s2

b. if (condition) then s1

Often the condition is called a ___because it is stationed before s1 and s2 and restricts access to them.

guard

____ used when a sequence of algorithm statements is to be executed over and over again.

Iterative statements

two types of iterative statements?

while loop
for-next

where the condition is a predicate involving algorithm variables. The word while marks the beginning of the loop, and the word end while mark its end.

while loop

A while loop has the form?

while (condition)
[statements that make up the body of the loop]
end while

A for-next loop has the following form:

for variable:= initial expression to final expression
[statements that make up
the body of the loop]

next (same) variable

This discussion can be summarized in a table and it shows the current values of algorithm variables at various points during execution.

Trace table -

The founder of set theory

Georg Cantor -

Cantor used the letter M because it is the first letter of the German word for set:?

Menge

The founder of set theory, Georg Cantor, suggested imagining a set as a “___________"

collection into a whole M of definite and separate objects of our intuition or our thought. These objects are called the elements of M.

A set may be specified using the ______ by writing all of its elements between braces.

set-roster notation

(The symbol ....... is called an ____ and is read "and so forth.")

ellipsis

The ____ says that a set is completely determined by what its elements are—not the order in which they might be listed or the fact that some elements might be listed more than once.

axiom of extension

Another way to specify a set uses what is called the___?

set-builder notation

It follows from the definition that for a set A not to be a subset of a set B means that there is at least one element of A that is not an element of B.

Subsets

A _____of a set is a subset that is not equal to its
containing set.

We have known that by the axiom of extension, sets A and B are equal if, and only if, they have exactly the same
Elements.

Set Equality

If sets A and B are represented as regions in the plane, relationships between A and B can be represented by pictures, called _____

Venn Diagrams

Venn Diagram was introduced by the British mathematician named _____ in ___ (year)

John Venn in 1881

A is a subset of B in venn diagram

A is not sub set of B in venn diagram

these operations are the union (∪), intersection(∩), complement
(′), difference (−), and the Cartesian or cross product (×).

Operations on Sets

The ___operation combines all elements of two sets. Any element that occurs in both sets only occurs once in the new set.

union (∪)

The union of A and B, denoted___, is the set of all elements that are in at least
one of A or B.

A U B

the intersection of A and B, denoted ___, is the set of all elements that are common to both A and B.

A ∩ B

The difference of B minus A (or relative complement of A in B), denoted ____
, is the set of all elements that are in B and not A.

B - A

The complement of A, denoted ___, is the set of all elements in U that are
whined

A‘

Venn diagram representations for union,

Venn diagram representations for intersection

Venn diagram representations for difference

Venn diagram representations for complement

what do you call to a set that has No element and denote it by the symbol __?

The Empty Set or null set

Ø

sets are divided up into nonoverlapping (or disjoint) pieces. Such a division is
called a ___.

partition

Two sets are called disjoint if, and only if, ______.

they have no elements in common

Sets A1, A2, A3... are ____(or pairwise disjoint or nonoverlapping)
if, and only if, no two sets Ai and Aj with distinct subscripts have any elements in
common.


mutually disjoint

The _____guarantees that this is a set.

power set axiom