Discrete Structures Week 2

Question 1

logical operators

Answer 1

produce new propositions out of old ones
- negation
- conjunction
-disconjunction
-exclusive or
-conditional
-biconditional
- equivalence

Question 2

logical equivalence

Answer 2

p and q are equivalent id they have the same truth table

Question 3

contrapositive

Answer 3

p -> q =(3lines) not q then not p

Question 4

p if q is a different way to say…

Answer 4

q then p

Question 5

p only if q means

Answer 5

p is T only if q is T so
not q -> not p = p -> q

Question 6

1st Morgan Law

Answer 6

not (p ^ q) = not p v not q

Question 7

every logical operator is equivalent to…

Answer 7

one that is written only in terms of ^, v, not

Question 8

two operators A and B are logically equivalents if and only if…

Answer 8

A is true whenever B is true and A is false whenever B is false

Question 9

2nd De Morgan Law

Answer 9

not (p v q) = not p ^ not q
- proof for this

Question 10

T is # and F is #
and how to solve these problems

Answer 10

T is 1, F is 0

• compare is corresponding number with the operator

Question 11

Contrapositive

Answer 11

p -> q = not q -> not p

Question 12

De Morgan Laws

Answer 12

not (p ^ q) = not p v not q
- prove by truth table

not (p v q) = not p ^ not q
- two propositions proof
p = q if P =T whenever q = T and p =F whenever q =F

Question 13

Tautology

Answer 13

a compound propositions that is always true
p v not p

Question 14

contradiction

Answer 14

a compound propositions that is always false
p ^ not p

Question 15

contingency

Answer 15

a compound proposition that is neither false or true always
p

Question 16

Argument

Answer 16

an assertion that gives some propositions P1…Pn (premises) imply another proposition q(conclusion)

Question 17

notation for arguments

Answer 17

P1…Pn |-> q

|-> is therefore

p1
p2
————-
three dots q

Question 18

valid

Answer 18

an argument p1…pn |-> q is valid (q=T) whenever p1…on are true

Question 19

fallacy

Answer 19

invalid
all premises (pn) are true but the conclusion(q) is false

Question 20

theorem of arguments

Answer 20

an argument p1…pn three dots is valid if and only if p1 ^p2^…^pn -> q is always true
- proof of this

Question 21

Modus Ponens

Answer 21

standard rules of inferences

p->q
p
——–
three dots q

Question 22

hypothetical syllogism

Answer 22

standard rules of inferences

p->q
q->r
———
three dots p->r