Flashcards in Module 2 Deck (34):

1

## What is predicate calculus?

### The symbolic analysis of predicates and quantified statements.

2

## What is statement calculus?

### The symbolic analysis of ordinary compound statements.

3

## What is a predicate?

### A sentence the contains a finite number of variables and becomes a statement when specific values are substituted for variables.

4

## What is the predicate domain?

### The set of all values that may be substituted in place of the variable.

5

## What is the truth set?

### The set of all elements that make the predicate true.

6

## How is the truth set denoted?

### {x E D | P(x)}

7

## What is a quantified?

### A word that refers to a quantity such as “some” or “all” and tell for how many elements a given predicate is true.

8

## What is the universal quantifier?

### Upside down A, means “for all”

9

## What can be used instead of “for all”?

###
“For arbitrary”

“For every”

“For any”

“For each”

“Given any”

10

## When are universal statements true?

### True if, and only if, it is true for every x in its domain.

11

## What is the method of exhaustion?

### Showing the truth of the predicate separately for each individual element in the domain.

12

## What is the existential qualifier?

### Backwards E, denotes “there exists”

13

## What can be used instead of “there exists”?

###
“There is a”

“We can find a”

“There is at least one”

“For some”

“For at least we one”

14

## When are existential statements true?

### True if, and only if, the predicate is true for at least one value in the domain.

15

## What form does a universal conditional statement have?

### Ax, if P(x) then Q(x)

16

## What is implicit quantification?

### P(x) => Q(x), every element in the truth set of P(x) is in Q(x).

17

## What does P(x) <=> Q(x) mean?

### P(x) and Q(x) have identical truth sets.

18

## What is the negation of a universal statement?

### It is logically equivalent to an existential statement saying there is one value that is the negation of the predicate.

19

## What is the negation of an existential statement?

### It is logically equivalent to a universal statement with the negation if the predicate.

20

## When is an integer even?

### If, and only if, n=2*(some integer)

21

## When is an integer odd?

### If, and only if, n=(2*(some integer))+1

22

## When is an integer prime?

### If, and only if: n>1, for all positive integers r and s, if n=rs, and r or s =n.

23

## When is an integer composite?

### If, and only if: n>1, for all positive integers r and s, n=rs with 1

24

## What is a constructive proof of existence?

### Find a value in the domain to prove the predicate is true or provide instructions for finding such a value.

25

## What is a nonconstructive proof of existence?

### Show either: (a) that the existence of a value of x that makes the predicate the predicate true is guaranteed by an axiom or theorem (b) that the assumption that there is no such x leads to a contradiction.

26

## What is a disproof by counterexample?

### To disprove a statement by finding a value of x for which the statement is false.

27

## What is the method of generalizing from the generic particular?

### 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.

28

## What is the method of direct proof?

###
1. Express statement in quantifiable terms.

2. Utilize method of generalizing from the generic particular.

3. Show conclusion is true by using definitions, previous results, and logical inference.

29

## What is existential instantiation?

### If the existence of a certain kind of object is assumed or has been deduced then it can be given a name, as long as that name is not currently being used to denote something else.

30

## What is a corollary?

### A statement whose truth can be immediately deduced from a theorem that has already been proven.

31

## What is the definition if divisibility?

### If n and d are integers and d =/=0, n is divisible by d if, and only if, n=d*(some integer)

32

## What are the two rules of divisibility?

###
1) if one positive integer divides another positive integer, then the first is less than or equal to the second.

2) only divisors of 1 are 1 and -1

33

## What is the method of proof by division if cases?

### If many cases are presented for the same conclusion, separating them out individually

34