Lecture 1 and 2

Question 1

How to convert Binary to Decimal format

Answer 1

To convert binary to decimal format, starting with the leftmost digit add together 1 x 2^n.
Example- 1110011 is equivalent to 115. 1x2^6 + 1 x 2^5 + 1 x 2^4 + 1 x 2^1 + 1 x 2 ^0. You exclude any that have a 0 in the slot.

Question 2

How to convert decimal to binary format?

Answer 2

To convert decimal to binary format you continually divide by 2 and track the remainder and then read the remainders backwards.

Question 3

How to convert decimal to hexadecimal?

Answer 3

To convert decimal to hexadecimal you divide the number by 16 and read the remainders backwards.

Question 4

How to convert binary to hexadecimal?

Answer 4

To convert binary to hexadecimal you split the binary into groups of 4 and if there is an uneven group of less than 4 add a 0 to the left hand side.

Question 5

How to convert binary into octal

Answer 5

To convert binary to octal you split the binary into group of 3

Question 6

How to express a decimal based fraction in binary?

Answer 6

Start by splitting the whole number portion and the fraction
Then multiply the fraction portion by 2 and record the 1 on a separate column and continue to only multiply the fraction portion.

Question 7

What is the double negative law?

Answer 7

~(~p) = p

Question 8

What is the commutative law

Answer 8

p v q = q v p

Question 9

What is the Associative Law?

Answer 9

(p v q) v r = p v (q v r) and

p ^ q) ^ r = p^ (q^r

Question 10

What is the distributive law?

Answer 10

p ^ (q v r) = (p^ q) v (p^r) and
p v (q^r) = (p v q) ^ (p v r)

Question 11

What is Demorgan’s Law

Answer 11

~(p v q) = ~p ^ ~q

~(p ^ q) = ~p v ~q

Question 12

Difference between predicate and proposition?

Answer 12

In contrast predicate logic distinguishes the subject of a sentence from its predicate.

i.e “The dog is sleeping”
the phrase is sleeping is the predicate and is symbolized by P(x)

Question 13

How do you rewrite p XOR q?

Answer 13

2 ways:

1. (p v q) ^ ~(p ^ q)
2. (p ^ ~q) v (q ^ ~p)

Question 14

How to rewrite p –> q

Answer 14

~p v q

Question 15

How to rewrite p q

Answer 15

2 Ways:

1. (p –> q) ^ (q –>p)
2. ~(p XOR q)

Question 16

What is the negation of: For all x, P(x)

Answer 16

~(For all x, P(x)) <=> There exists x, ~p(x)

This works for there exists statements as well. to negate these statements change ‘for all’ to ‘there exists’ and vice versa

Question 17

What is a Theorem (proposition)?

Answer 17

a statement that has been proven to be true.

Question 18

What are Axioms, postulates, hypotheses, premises?

Answer 18

They are assumptions (often unproven) defining the structures about which we are reasoning.

Question 19

What are Rules of inference

Answer 19

patterns of logically valid deductions from hypotheses to conclusions.

Question 20

What is a Lemma

Answer 20

a minor theorem used as a stepping stone to proving a major theorem.

Question 21

What is a corollary?

Answer 21

a minor theorem proved as an easy consequence of a major theorem

Question 22

What is a conjecture?

Answer 22

a statement whose truth value has not been proven. May be widely believed to be true anyway.

Question 23

What is a theory?

Answer 23

the set of all theorems that can be proven.

Question 24

What is the rule of Addition pertaining to Inference Rules

Answer 24

p is true

p v q is true

Question 25

What is the rule of simplification pertaining to inference rules

Answer 25

p ^ q is true

q is true

Question 26

What is the rule of conjunction pertaining to inference rules?

Answer 26

p is true
q is true
p ^ q is true

Question 27

What is the rule of Modus Ponens aka Law of detachment

Answer 27

p –> q
p is true
therefore q

Question 28

What is the rule of Modus tollens aka the mode of denying?

Answer 28

p implies q
not p
therefore not q

Question 29

What is the rule of hypothetical syllogism

Answer 29

p implies q
q implies r
therefore p implies r

Question 30

What is the rule of disjunctive syllogism?

Answer 30

p v q
not p
therefore q

Question 31

What is Universal instantiation?

Answer 31

For all x P(x)
therefore P(o) (some value)

Question 32

What is Universal generalization?

Answer 32

P(g) of some arbitrary g

therefore for all x P(x)

Question 33

What is existential instantiation

Answer 33

there exists x such that P(x)
therefore P(c)

Question 34

What is existential generalization?

Answer 34

P(o)

therefore there exists x such that P(x)

Question 35

Proof Methods for implications (p implies q) 1. Direct Proof

Answer 35

assume p is true and prove q

Question 36

Proof methods for implications 2. Indirect Proof

Answer 36

Assume ~q and prove ~p

Question 37

Proof methods for implications 3. Vacuous Proof

Answer 37

prove ~p by itself

Question 38

Proof methods for implications 4. Trivial proof

Answer 38

prove q by itself

Question 39

What is (k+1)! equal to?

Answer 39

(k+1)*k!