SECTION 1.4
Equivalent Statements 31
Textbook Reference Section 3.3, 3.4
1.4 EQUIVALENT STATEMENTS
CLAST OBJECTIVE
" Determine equivalent and non-equivalent statements
Equivalent Statements are statements that are written differently, but hold the same
logical equivalence.
Case 1: “ If p then q ” has three equivalent statements.
Statement
RULE
Equivalent Statement
1) If p then q
not p or q
2) If p then q
q or not p
3) If p then q
If not q then not p
Examples
Statement
a) If it is blue, then it is the sky.
Equivalent Statement
It is not blue or it is the sky.
b) If it is blue, then it is the sky.
It is the sky or it is not blue.
c) If it is blue, then it is the sky.
If it is not the sky, then it is not blue.
Case 2: Statements beginning with “ It is not true that… ”
RULE
Statement
1) ~ (p and q)
Equivalent Statement
not p or not q
2)
3)
4)
5)
not p and not q
p and not q
Some p are not q
Some p are q
~ (p or q)
~ (if p then q)
~ (All p are q)
~ (No p are q)
6) ~ (Some p are q)
7) ~ (Some p are not q)
No p are q
All p are q
Examples
Statement
Equivalent Statement
d) It is not true that Peter has an A and
Mary has a C.
Peter does not have a A or Mary does not
have a C.
e) It is not true that the car is red or the
pole is not blue.
The car is not red and the pole is blue.
32 CHAPTER 1
Sets and Logic
Examples
Statement
Equivalent Statement
f) It is not true that if Peter has an A, then
Mary has a C.
Peter has an A and Mary does not have a C.
g) It is not true that all apples are red.
Some apples are not red.
h) It is not true that no banana is yellow.
Some bananas are yellow.
i) It is not true that some grapes are not
green.
All grapes are green.
Check Your Progress 1.4
For Questions 1- 4, answer “ yes ” if the statements are equivalent. Answer “ no ” if the
statements are not equivalent.
1. If it is heavy, then I am strong.
If I am not strong, then it is not heavy.
2. If it is winter, then Tom is cold.
It is winter and Tom is cold.
3. If Jack wins, then Mary loses.
Jack does not win or Mary loses.
4. If cats like fish, the dogs like beef.
Dogs like beef or cats do not like fish.
For Questions 5 – 9, write an equivalent statement.
5. It is not true that Ben plays the flute or Jan plays the piano.
6. It is not true that chocolate is white and milk is green.
7. It is not true that all puppies are playful.
8. It is not true that some elephants are grey.
9. It is not true that no berries are red.
SECTION 1.4
See If You
Remember
Equivalent Statements 33
SECTIONS 1.1 – 1.3
For Questions 1 and 2, write the rule that directly transforms statement i) into statement
ii).
1.
i)
ii)
Not all clowns are funny.
Some clowns are not funny.
2.
i)
ii)
If Mary has a B, then Kyle has a D.
If Kyle does not have a D, then Mary does not have a B.
3.
i)
ii)
It is not true that you study French or English.
You do not study French and you do not study English.
For Questions 4 – 8, negate the statement.
4. No cats are furry.
5. All dogs are fluffy.
6. If Jim wins, then he flies to Vegas.
7. Tom is in French class and Ben is in Spanish class.
8. I am a student or Jean is a teacher.