1.4 EQUIVALENT STATEMENTS Textbook Reference Section 3.3, 3.4 CLAST ...

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 ¨C 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 ¨C 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 ¨C 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.

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