Counterexamples

Key Notes:

🎈 A conditional statement can be expressed as If A, then B.

🎈 A is the hypothesis and B is the conclusion.

🎈 A counterexample is an example in which the hypothesis is true, but the conclusion is false. If you can find a counterexample to a conditional statement, then that conditional statement is false.

Learn with an example

🛬 Which of the following is a counter

example for this conditional statement?

🛬 If a child is a girl, then the child’s favourite colour is pink. 

  • Ben, a boy whose favourite colour is pink
  • Kevin, a boy whose favourite colour is blue
  • Maura, a girl whose favourite colour is pink
  • Audrey, a girl whose favourite colour is green
  • First notice that the given statement is in if/then form:
If a child is a girl, then the child’s favourite colour is pink. 
If A, then B. 
  • Now go through the answer choices one by one, looking for the counterexample that makes the hypothesis true and the conclusion false.
  • First try Ben, a boy whose favourite colour is pink. The hypothesis is false, because this child is not a girl. A counterexample must make the hypothesis true, so this is not a counterexample.
  • Next try Kevin, a boy whose favourite colour is blue. The hypothesis is false, because this child is not a girl. A counterexample must make the hypothesis true, so this is not a counterexample.
  • Next try Maura, a girl whose favourite colour is pink. The hypothesis is true, because this child is a girl. The conclusion is also true, because this child’s favourite colour is pink. A counterexample must make the conclusion false, so this is not a counterexample.
  • Finally try Audrey, a girl whose favourite colour is green. The hypothesis is true, because this child is a girl. The conclusion is false, because this child’s favourite colour is not pink. A counterexample makes the hypothesis true and the conclusion false. So, this is a counterexample.
  • In summary, the counterexample is Audrey, a girl whose favourite colour is green.

🛬 Which of the following is a counterexample for this conditional statement?

🛬 If the sum of two integers is even, then those integers are both even. 

  • 8 and 1
  • 2 and 4
  • 6 and 4
  • 1 and 3
  • First notice that the given statement is in if/then form:
If the sum of two integers is even, then those integers are both even. 
If A, then B. 
  • Now go through the answer choices one by one, looking for the counterexample that makes the hypothesis true and the conclusion false.
  • First try 8 and 1. The hypothesis is false, because the sum of 8 and 1 is not even. A counterexample must make the hypothesis true, so this is not a counterexample.
  • Next try 2 and 4. The hypothesis is true, because the sum of 2 and 4 is even. The conclusion is also true, because 2 and 4 are both even. A counterexample must make the conclusion false, so this is not a counterexample.
  • Next try 6 and 4. The hypothesis is true, because the sum of 6 and 4 is even. The conclusion is also true, because 6 and 4 are both even. A counterexample must make the conclusion false, so this is not a counterexample.
  • Finally try 1 and 3. The hypothesis is true, because the sum of 1 and 3 is even. The conclusion is false, because 1 and 3 are not both even. A counterexample makes the hypothesis true and the conclusion false. So, this is a counterexample.
  • In summary, the counterexample is 1 and 3.

🛬Which of the following is a counterexample for this conditional statement?

🛬 If an animal walks on two legs, then it is a human. 

  • an elephant
  • a giraffe
  • a kangaroo
  • a dolphin
  • First notice that the given statement is in if/then form:
If an animal walks on two legs, then it is a human. 
If A, then B. 
  • Now go through the answer choices one by one, looking for the counterexample that makes the hypothesis true and the conclusion false.
  • First try an elephant. The hypothesis is false, because an elephant does not walk on two legs. A counterexample must make the hypothesis true, so this is not a counterexample.
  • Next try a giraffe. The hypothesis is false, because a giraffe does not walk on two legs. A counterexample must make the hypothesis true, so this is not a counterexample.
  • Next try a kangaroo. The hypothesis is true, because a kangaroo walks on two legs. The conclusion is false, because a kangaroo is not a human. A counterexample makes the hypothesis true and the conclusion false. So, this is a counterexample.
  • Finally try a dolphin. The hypothesis is false, because a dolphin does not walk on two legs. A counterexample must make the hypothesis true, so this is not a counterexample.
  • In summary, the counterexample is a kangaroo.

Let’s practice!🖊️