Equivalent Statements



Equivalent Statements

Prerequisite: The negation of a sentence involves putting in a “not” or taking it out.

Original Sentence Negation of this Sentence

The sky is blue. The sky is not blue.

This house is not new. This house is new.

P ~p

~q q

Three conditional statements are defined from an original sentence:

Original Statement p → q If the sky is blue, then it is not raining.

Converse of the Original q → p If it is not raining, then the sky is blue.

Inverse of the Original ~p → ~q If the sky is not blue, then it is raining.

Contrapositive of the Original ~q → ~p If it is raining, then the sky is not blue.

Match each of these four conditionals with its Venn diagram.

Figure 1 Figure 2 Figure 3 Figure 4

The original sentence matches figure _______.

The converse of the original matches figure _____.

The inverse of the original matches figure _____. (Where does being outside the circle representing p guarantee being outside the circle represent q?)

The contrapositive of the original matches figure _____. (Where does being outside the circle representing q guarantee being outside the circle representing p?)

The two conditionals represented by the same figure are said to be equivalent statements. Figure 1 represents the original conditional statement and the contrapositive of the original.

Key Point: The original conditional statement and the contrapositive of the original are logically equivalent statements and may be used interchangeably.

p → q is logically equivalent to ~q → ~p.

p → q may be replaced by ~q → ~p.

~q → ~p may be replaced by p → q.

Key Point: When the conditional, p → q, and its converse, q → p, are both true, then they may be combined in one statement called a biconditional statement, p ↔ q.

p ↔ q is read “p if and only if q”

p if q is the q → p (if q, then p) part so p only if q is the p → q (if p, then q) part. (“only if” means “then.”)

Exercises

I. Write the original statement in “if-then” form. Then write its converse, inverse, and contrapositive.

Example:

Original: An odd number cannot be an even number.

If-then form of the original: If a number is an odd number, then it cannot be an even number.

Converse of the original: If a number is not an even number, then it is an odd number.

Inverse of the original: If a number is not an odd number, then it can be an even number.

Contrapositive of the original: If a number is an even number, then it is not an odd number.

1. Original: All right angles are congruent.

If-then form of the original: ______________________________________________________

Converse of the original: _________________________________________________________

Inverse of the original: __________________________________________________________

Contrapositive of the original: _____________________________________________________

2. Original: Every equilateral triangle is also an isosceles triangle.

If-then form of the original: ______________________________________________________

Converse of the original: _________________________________________________________

Inverse of the original: __________________________________________________________

Contrapositive of the original: _____________________________________________________

3. Original: ~s → ~r

If-then form of the original: ______________________________________________________

Converse of the original: _________________________________________________________

Inverse of the original: __________________________________________________________

Contrapositive of the original: _____________________________________________________

4. Original: a → ~b

If-then form of the original: ______________________________________________________

Converse of the original: _________________________________________________________

Inverse of the original: __________________________________________________________

Contrapositive of the original: _____________________________________________________

II. Following each of the numbered statements below are three lettered statements. Identify the relationship of each of the lettered statements to the numbered statement. Write “original”, “converse”, “inverse”, “contrapositive”, or “none” as appropriate. Hint: Write each statement in “if-then” form.

5. If you live in Atlantis, then you need a snorkel.

___________________a) If you do not live in Atlantis, then you do not need a snorkel.

___________________b) If you need a snorkel, then you live in Atlantis.

___________________c) If you do not need a snorkel, then you do not live in Atlantis.

6. If you are over ninety, the Chop Chop Studio will give you free karate lessons.

___________________a) If the Shop Chop Studio won’t give you free karate lessons, then you aren’t over ninety.

___________________b) If you are ninety or less, the Chop Chop Studio will give you free karate lessons.

___________________c) The Chop Chop Studio will give you free karate lessons if you are over ninety.

7. All children like pie.

___________________a) If someone likes pie, then he is a child.

___________________b) If someone is not a child, he likes pie.

___________________c) A person who does not like pie is not a child.

8. Lady kangaroos do not need handbags.

___________________a) If a kangaroo is not a lady, it needs a handbag.

___________________b) If it needs a handbag, then it is not a lady kangaroo.

___________________c) A kangaroo does not need a handbag if it is a lady.

9. Terry will study if his brother takes him to the library.

___________________a) If his brother takes him to the library, then Terry will study.

___________________b) Because Terry’s brother is not taking him to the library, he will not study.

___________________c) Terry does not study in the library.

10. Terry will study only if his brother takes him to the library.

___________________a) If his brother takes him to the library, then Terry will study.

___________________b) Because Terry’s brother is not taking him to the library, he will not study.

___________________c) Terry does not study in the library.

11. Because teachers are fair, students get the grades they deserve.

___________________a) Because students get the grades they deserve, their teachers are fair.

___________________b) If students don’t get the grades they deserve, then their teachers are not fair.

___________________c) If teachers are fair, then students get the grades they deserve.

12. Tim will make the bus if he hurries.

___________________a) If Tim hurries, then he will make the bus.

___________________b) Tim did not hurry so he missed the bus.

___________________c) Tim refuses to hurry.

13. Nancy has lunch only if her mother makes it for her.

___________________a) If Nancy’s mother doesn’t make lunch, then Nancy doesn’t have lunch.

___________________b) If Nancy has lunch then her mother makes it for her.

___________________c) Nancy doesn’t buy lunch at the cafeteria.

14. Students who go to Clements High School are fun to teach.

___________________a) If a student is fun to teach, then he goes to Clements High School.

___________________b) If a student doesn’t go to Clements High School, then he is not fun to teach.

___________________c) Students have fun at Clements High School.

15. ~r → s

___________________a) r → s

___________________b) s → ~r

___________________c) ~s → r

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