1. Discrete Mathematics
Lecture # 6

2. Standard Connectives
connective
symbol
and  or 
not ~
if – then 

3. Non standard Connectives
… but …
… although …
… or … but not both
neither … nor …
… if …
… only if …
… if and only if …
… unless …
… is necessary for …
… is sufficient for …
… if … otherwise …
…unless…in which case

4. Conjunction
There standard English expression for conjunction is AND but there are other numerous conjunction-like expressions as follows:
But
Yet
Although
Though
Even though
Moreover
Furthermore
However
Whereas

5. Examples It is raining but I am happy.
Although it is raining, I am happy.
It is raining yet I am happy.
It is raining and I am happy.

6. Disguised Conjunctions
Other examples of disguised conjunctions involved relative pronoun are:
Who
Which
That
For example
Jones is a former player who coaches basket ball.
Jones is a former player and Jones coaches basket ball.

7. Sometimes ‘and’ is used as follows. keep trying, and you will succeed
Odd use of ‘and’
Sometimes ‘and’ is used as follows.
keep trying, and you will succeed
keep it up buster, and I will clobber you
give him an inch, and he will take a mile
give me the money, and I’ll give you the product
you do that, and I’ll kill you
Truth table for (pq)  ~ (pq)

8. Negations For instance: It is not true that _________.
It is false that _____________.
It is not the case that ___________.

9. Exclusive OR Ram xor Sam will win (the election)
Ram will win, or Sam will win, but not both
at least one of them win
( R  S )
also
&
both of them will win…not!
~( R ^ S )
alternatively,
Ram will win, and Sam will not win,
( R ^ ~ S ) or 
Sam will win, and Ram will not win.
( S ^ ~R )

10. Neither…Nor it is neither raining nor sleeting
it is not raining, and it is not sleeting
~ R ^ ~ S
neither Jay nor Kay is sleeping
Jay is not sleeping, and Kay is not sleeping
~ J ^ ~ K

11. Alternative Formulation
it is neither raining nor sleeting
it is not raining or sleeting
it is not true that ( it is raining or it is sleeting )
~ ( R  S ) 
~ R ^ ~S

12. A  C Conditionals standard expression: if A then C
which is symbolized:
A  C
A is the antecedent
C is the consequent

13. A  C Variants of if…then if A, then C if A, C C if A all of these are
symbolized the
same:
A  C
‘ if ’ always introduces the antecedent
‘ then ’ always introduces the consequent

14. all symbolized the same way
Other variants of ‘if’
C provided A
C in case A
C supposing A
provided A, C
in case A, C
supposing A, C
all symbolized the same way
A  C

15. Only if ‘ only if ’ is not equivalent to ‘ if ’ T F T F
I will get an A only if I take all the exams
versus
I will get an A if I take all the exams T F
I will get an A if I get a hundred
versus
I will get an A only if I get a hundred T F

16. ‘only’ operates as a dual-negative modifier.
How does only works?
employees only
authorized personnel only
cars only
right turn only
‘only’ operates as a dual-negative modifier.
for example, (1) means to
exclude anyone who is
not an employee.

17. How does ‘only’ modifies ‘if’
by introducing two negations
A only if B
not A if not B
if not B
then
not A
~ B  ~ A

18. Example A only if E not A if not E if not E then not A ~ E  ~ A
I will get an A only if I take all the exams
A only if E
I will not get an A if I do not take all the exams
not A if not E
if I do not take all the exams then I will not get an A
if not E then not A
~ E  ~ A

19. Unless unless = if not The following are all equivalent.
I will pass only if I study P only if S
I will not pass unless I study not P unless S
I will not pass if I do not study not P if not S
only if has two built-in negations
unless has one built-in negation
if has no built-in negation
unless = if not

20. I will not pass unless I study
Example
I will not pass unless I study
not P
unless S
not P
if not S
not P if
not S
if not S
then
not P
~ S 
~ P

21. I will play tennis if it is sunny; otherwise, I will play racket ball
If…otherwise
I will play tennis if it is sunny; otherwise, I will play racket ball
T if S ; otherwise, R
this answers two questions:
what will I do IF it is sunny?
play tennis
what will I do OTHERWISE (i.e., IF it is NOT sunny)?
play racket ball

22. If…otherwise if it’s sunny, then I’ll play tennis furthermore
if it’s not sunny, then I’ll play racketball
if S then T
and
if not S then R
( S  T ) 
(~ S  R )

23. Unless…in which case T unless R, in which case S
I will play tennis unless it rains, in which case I will play squash
T unless R, in which case S
this answers two questions:
what will I do unless it rains
(i.e., if it does not rain)?
play tennis
what will I do in case it rains
(i.e., if it does rain)?
play squash

24. Unless…in which case if it does not rain, then I’ll play tennis
furthermore
if it does rain, then I’ll play squash
if not-R then T
and
if R then S
(~R  T ) 
( R  S )

25. Necessary Conditions in order that I get an A it is necessary that
I take 4 exams
in order for
me to get an A
it is necessary for
me to take 4 exams
in order
to get an A
it is necessary
to take 4 exams
taking 4 exams
is necessary for
getting an A

26. Necessary Conditions if not A then not B ~A  ~B simplest paraphrase:
A is necessary for B
this amounts to saying
if A does not happen,
then neither does B
if not A then not B
~A  ~B

27. Example if not E then not A ~E  ~A
taking 4 exams E is necessary for getting an A A
if E does not happen,
then neither does A
if not E then not A
~E  ~A

28. Sufficient Conditions
in order that
I get an A
it is sufficient that
I get a hundred
in order for
me to get an A
it is sufficient for
me to get a hundred
in order
to get an A
it is sufficient
to get a hundred
getting a hundred
is sufficient for
getting an A

29. Sufficient Conditions
simplest paraphrase:
A is sufficient for B
this amounts to saying
if A does happen,
then so does B
if A then B
A  B

30. Example
getting a hundred H is sufficient for getting an A A
if H does happen,
then so does A
if H then A
H  A

31. Negation of Necessity ( ) if then not H is nec for A if not H then
getting a hundred H is NOT necessary for getting an A A
not H
is nec for A if
not H
then
not A ~ (
~ H 
~ A )
~ (~ H  ~ A )
it is not true that if
you don’t get a H
then
you won’t get an A

32. Negation of Sufficiency
Taking all the exams E is NOT sufficient for getting an A A
not E
is suf for A if E
then A ~ ( E  A )
~ (E  A )
it is not true that if
you (merely) take the exams
then
you will get an A

33. Basic Statements A is necessary for B A is sufficient for B
A is not necessary for B
A is not sufficient for B

34. Combinations A is both necessary and sufficient for B
A is necessary, but not sufficient, for B
A is sufficient, but not necessary, for B
A is neither necessary nor sufficient for B

35. Example 1 (~ A  ~ P ) ( A  P ) averaging (at least) fifty A
is both necessary and sufficient for
passing P
A is necessary for P
(~ A  ~ P )
and 
A is sufficient for P
( A  P )

36. E is not sufficient for A
Example 2
taking four exams E
is necessary but not sufficient for
getting an A A
E is necessary for A
(~E  ~A )
but 
E is not sufficient for A
~ ( E  A )

37. Example 3 ( H  A )  getting a hundred H
is sufficient but not necessary for
getting an A A
H is sufficient for A
( H  A )
but 
E is not necessary for A
~ (~H  ~A )

38. A is not sufficient for P
Example 4
attending class A
is neither necessary nor sufficient for
passing P
A is not necessary for P
~ (~ A  ~ P )
and 
A is not sufficient for P
~ ( A  P )

39. Complex Statements (Step by Step Procedure)
1. Identify the simple (atomic) statements; abbreviate them by upper case letters.
2. Identify all the connectives. Which are standard? Which are non-standard?
3. Rewrite the sentence replacing the simple statements by their abbreviations. Retain internal punctuation.
4. Identity the major connective. [Notice commas!]
5. If the major connective is standard, symbolize it; otherwise, paraphrase it and repeat 4.
6. Work on the constituent formulas resulting from 5 [go to 4].
7. Substitute constituents back into overall formula.
8. Translate the formula back into English, and compare it with original sentence.

40. Example 1
if neither JAY nor KAY is working, then we will go on VACATION.
1. simple sentences
J = Jay is working
K = Kay is working
V = we go on vacation
2. connectives
if...then (standard)
neither...nor (non-standard)

41. Example 1(contd.) ( neither J nor K )  V 3. first formula
if neither J nor K, then V
4. main connective?
5. standard or non-standard?
standard, so we symbolize it
( neither J nor K )  V

42. Example 1(contd.) ( not J and not K ) (~ J  ~ K )
6. work on constituents
( neither J nor K )  V
( not J and not K )
(~ J  ~ K )
7. substitute constituents
(~ J  ~ K )  V
8. translate formula back into English
if Jay is not working, and Kay is not working,
then we will go on vacation.

43. Example 2 unless the exam is EASY, I will PASS only if I STUDY
1. simple sentences
E = the exam is easy
P = I pass
S = I study
2. connectives
unless (non-standard)
only if (non-standard)

44. Example 2(contd.) 3. first formula unless E, P only if S
4. main connective
5. standard or non-standard
non-standard, so we paraphrase it
unless = if not
if not E, then P only if S

45. Example 2(contd.) if not E, then P only if S 4. main connective
5. standard or non-standard
standard, so we symbolize it
( not E )  ( P only if S )
6. work on constituents
( not E )  ( P only if S )
~ E
( not P if not S )
( if not S, then not P )
(~ S  ~ P )

46. Example 2 (contd.) 7. Substitute constituents ~ E  (~ S  ~ P )
8. Translate formula back into English if
the exam is not easy,
then
I do not study,
I will not pass

47. Example 3
if I am HAPPY only if I am DRUNK, then I am not HAPPY unless I am DRUNK.
1. simple sentences:
H = I am happy
D = I am drunk
2. connectives:
if…then (standard)
only if (non-standard)
not (standard)
unless (non-standard)

48. Example 3 (contd.)
3. first formula: if H only if D, then not-H unless D
4. main connective:
if H only if D, then not-H unless D
5. standard or non-standard:
standard, so we symbolize it
( H only if D )  ( not-H unless D )

49. Example 3 (contd.) 6. Work on constituents
( H only if D )  ( not-H unless D )
( not H if not D )
(if not D, then not H )
(~ D  ~ H )
( H only if D )  ( not-H unless D )
( not-H if not D )
( if not-D then not H )

50. Example 3 (contd.) 7. Substitute constituents
(~ D  ~ H )  (~ D  ~ H )
8. Translate formula back into English IF
if I am not drunk
then I am not happy
THEN

51. Example 4 if I CONCENTRATE well only if I am ALERT,
then provided I am WISE
I will not FLY an airplane unless I am SOBER
1. simple sentences:
C = I concentrate well
A = I am alert
W = I am wise
F = I fly an airplane
S = I am sober
2. connectives:
if…then (standard)
only if (non-standard)
provided (non-standard)
not (standard)
unless (non-standard)

52. Example 4 (contd.) 3. first formula:
if C only if A, then provided W, not F unless S
4. main connective:
if C only if A, then provided W not F unless S
5. standard or non-standard
standard, so we symbolize it
( C only if A )  ( provided W, not F unless S )

53. Example 4 (contd.) 6a. work on first constituent – antecedent
( C only if A )  ( provided W, not-F unless S )
( not C if not A )
(if not A, then not C )
(~ A  ~ C )

54. Example 4 (contd.) 6b. work on second constituent – consequent
( C only if A )  (provided W, not-F unless S)
main connective?
provided W, not-F unless S
paraphrase?
if W, then ( not-F unless S )
W  ( not F unless S )
W  ( not F if not S )
W  ( if not S, then not F )
W  (~ S  ~ F )

55. Example 4 (contd.) substitute constituents
(~ A  ~ C)  ( W  (~ S  ~ F) )
8. Translate formula back into English IF
if I am not alert
then I do not concentrate well
THEN
if I am wise,
then
if I am not sober
then I will not fly an airplane