Equivalencies

If A, then B is equivalent to If not B, then not A

Not (A and B) is equivalent to not A OR not B

Not (A or B) is equivalent to not A AND not B

Can you give an equivalent statement to
1. If an Iowa farmer does not grow corn or beans, then he grows alfalfa.

2. It is not true that both Baghdad and Iraq are countries.

3. If my nephew is playing BB tonight, then I am going to the game.

4. If it is cold in Iowa, then I turn up the thermostat.

5. If I crack black walnuts for the bird feeder, then many birds come back often to feed.

6. If the sun does not shine, then Jane does not draw the blinds shut.

1. An equivalent statement to "If an Iowa farmer does not grow corn or beans, then he grows alfalfa" would be "If an Iowa farmer does not grow alfalfa, then he grows corn or beans."

Explanation: We can apply the contrapositive of the original statement. The contrapositive of "If A, then B" is "If not B, then not A." In the original statement, A represents not growing corn or beans, and B represents growing alfalfa. So, the contrapositive statement becomes "If not growing alfalfa, then growing corn or beans."

2. An equivalent statement to "It is not true that both Baghdad and Iraq are countries" would be "Either Baghdad is not a country or Iraq is not a country."

Explanation: We can apply De Morgan's laws to the negation of the original statement. The negation of "A and B" is "not A or not B." In the original statement, A represents Baghdad being a country, and B represents Iraq being a country. So, applying De Morgan's laws, we get "Either Baghdad is not a country or Iraq is not a country."

3. An equivalent statement to "If my nephew is playing BB tonight, then I am going to the game" would be "If I am not going to the game, then my nephew is not playing BB tonight."

Explanation: We can apply the contrapositive of the original statement. The contrapositive of "If A, then B" is "If not B, then not A." In the original statement, A represents my nephew playing BB tonight, and B represents me going to the game. So, the contrapositive statement becomes "If I am not going to the game, then my nephew is not playing BB tonight."

4. An equivalent statement to "If it is cold in Iowa, then I turn up the thermostat" would be "If I do not turn up the thermostat, then it is not cold in Iowa."

Explanation: We can apply the contrapositive of the original statement. The contrapositive of "If A, then B" is "If not B, then not A." In the original statement, A represents it being cold in Iowa, and B represents me turning up the thermostat. So, the contrapositive statement becomes "If I do not turn up the thermostat, then it is not cold in Iowa."

5. An equivalent statement to "If I crack black walnuts for the bird feeder, then many birds come back often to feed" would be "If many birds do not come back often to feed, then I do not crack black walnuts for the bird feeder."

Explanation: We can apply the contrapositive of the original statement. The contrapositive of "If A, then B" is "If not B, then not A." In the original statement, A represents cracking black walnuts for the bird feeder, and B represents many birds coming back often to feed. So, the contrapositive statement becomes "If many birds do not come back often to feed, then I do not crack black walnuts for the bird feeder."

6. An equivalent statement to "If the sun does not shine, then Jane does not draw the blinds shut" would be "If Jane draws the blinds shut, then the sun shines."

Explanation: We can apply the contrapositive of the original statement. The contrapositive of "If A, then B" is "If not B, then not A." In the original statement, A represents the sun not shining, and B represents Jane drawing the blinds shut. So, the contrapositive statement becomes "If Jane draws the blinds shut, then the sun shines."

It would be an equivalent statement