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5.5. DeMorgan’s Laws

DeMorgan’s laws were developed by Augustus De Morgan in the 1800s. They show how to handle the negation of a complex conditional, which is a conditional statement with more than one condition joined by an and (&&) or or (||), such as (x < 3) && (y > 2).

  • not (a and b) is the same as (not a) or (not b). In Java this is written as !(a && b) == !a || !b

  • not (a or b) is the same as (not a) and (not b). In Java this is written as !(a || b) == !a && !b

Applying DeMorgan’s Laws to !(x < 3 && y > 2) yields !(x < 3) || !(y > 2) which means that this complex conditional will be true when (x >= 3 || y <= 2).

Note

Notice that the negation is distributed to all parts of a complex conditional. It negates each part of the conditional and changes and (&&) to or (||) and or (||) to and (&&).

The negation modifies each conditional as shown below.

  • < becomes >=

  • > becomes <=

  • == becomes !=

  • <= becomes >

  • >= becomes <

  • != becomes ==

See the example below. For what values of x and y will it print true? Try out different values of x and y to check your answer.

For more information about DeMorgan’s laws see http://en.wikipedia.org/wiki/De_Morgan’s_laws.

    5-5-1: What is printed when the following code executes and x equals 4 and y equals 3?

    if (!(x < 3 || y > 2)) System.out.println("first case");
    else System.out.println("second case");
    
  • first case
  • This will be printed if x is greater or equal to 3 and y is less than or equal to 2. The first part is true but the second is false. Since the statements are joined by an and the complex conditional is false.
  • second case
  • This will be printed if x is less than 3 or y is greater than 2. In this case the first will be false, but the second true so since the statements are joined with an or the complex conditional is true.

    5-5-2: What is printed when the following code executes and x equals 4 and y equals 3?

    if (!(x < 3 && y > 2)) System.out.println("first case");
    else System.out.println("second case");
    
  • first case
  • This will be printed if x is greater than or equal to 3 or y is less than or equal to 2. In this case x is greater than 3 so the first condition is true.
  • second case
  • This will be printed if x is less than 3 and y is greater than 2.

    5-5-3: Which of the following is the same as the code below?

    !(x > 2 && y < 4)
    
  • (x < 2) || (y > 4)
  • The negation of x > 2 is x <= 2
  • (x < 2) && (y > 4)
  • Don't forget that the and is changed to an or
  • (x <= 2) || (y >= 4)
  • The x > 2 becomes x <= 2, the y < 4 becomes y >= 4 and the and changes to or
  • (x <= 2) && (y >= 4)
  • Don't forget that the and is changed to an or

    5-5-4: Which of the following is the same as the code below?

    !(x == 2 && y > 4)
    
  • (x != 2) || (y < 4)
  • The negation of y > 4 is y <= 4
  • (x != 2) && (y < 4)
  • Don't forget that the and is changed to an or
  • (x != 2) && (y <= 4)
  • Don't forget that the and is changed to an or
  • (x != 2) || (y <= 4)
  • The and is changed to an or, the (x == 2) becomes (x != 2) and (y > 4) becomes (y <= 4)

    5-5-5: Which of the following is the same as the code below?

    !(x!=5 && y!=7)
    
  • (x == 5) || (y == 7)
  • The negation of && is || and the negation of != is ==
  • (x == 5) && (y == 7)
  • The negation of && is ||
  • (x != 5) || (y != 7)
  • The negation of x != 5 is x == 5. The negation of y != 7 is y == 7.
  • (x < 5) || (x > 5) || (y > 7) || (y < 7)
  • The negation of == is != which is the same as < or >. The negation of != is ==.

    5-5-6: Which of the following is the same as the code below?

    !(x<= 5 && y > 7)
    
  • (x > 5) && (y < 7)
  • The negation of && is || and the negation of y > 7 is y <= 7.
  • (x > 5) || (y < 7)
  • The negation of y > 7 is y <= 7.
  • (x > 5) && (y <= 7)
  • The negation of && is ||.
  • (x > 5) || (y <= 7)
  • The negation of (x <= 5) is (x > 5). The negation of && is ||. The negation of (y > 7) is (y <= 7).
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