2.6 Comparing Boolean Expressions

1. Equivalent expressions and truth tables

Two Boolean expressions are equivalent if they evaluate to the same value in all cases. A truth table can prove they are equivalent 2.6.A.1: list every combination of true and false for the variables, evaluate both expressions in each row, and compare. With two variables there are 4 rows; with three variables there are 8.

Are !(a && b) and !a || !b equivalent?

a	b	a && b	!(a && b)	!a	!b	!a || !b
true	true	true	false	false	false	false
true	false	false	true	false	true	true
false	true	false	true	true	false	true
false	false	false	true	true	true	true

The columns for !(a && b) and !a || !b match in every row, so the expressions are equivalent.

Are !(a && b) and !a && !b equivalent? Row 2 (a true, b false) gives true for the first and false for the second. One row that differs is enough to show two expressions are not equivalent.

boolean a = true;
boolean b = false;
System.out.println(!(a && b));
System.out.println(!a || !b);
System.out.println(!a && !b);

Output:

true
true
false

2. De Morgan’s laws

De Morgan’s laws can be applied to Boolean expressions to create equivalent Boolean expressions 2.6.A.2:

• !(a && b) is equivalent to !a || !b
• !(a || b) is equivalent to !a && !b

To apply a law: move the ! onto each part, and switch && to || (or || to &&). When the parts are comparisons, negate each comparison:

Comparison	Its negation
x > 5	x <= 5
x < 5	x >= 5
x >= 5	x < 5
x == 5	x != 5

Example: !(temp > 30 || isRaining) becomes temp <= 30 && !isRaining. In words: “not (hot or raining)” means “not hot and not raining”.

int temp = 24;
boolean isRaining = false;
boolean original = !(temp > 30 || isRaining);
boolean rewritten = temp <= 30 && !isRaining;
System.out.println(original + " " + rewritten);

Output:

true true

Watch out: a common wrong answer keeps the > and writes temp < 30. For temp equal to 30, !(temp > 30) is true but temp < 30 is false.

3. Two variables, one object

Two different variables can hold references to the same object. Object references can be compared using == and != 2.6.B.1. The comparison is true only when both references point to the same object; it does not look at what is inside the objects.

String teamA = new String("Lions");
String teamB = new String("Lions");
String favorite = teamA;
System.out.println(teamA == favorite);
System.out.println(teamA == teamB);
System.out.println(teamA != teamB);

Output:

true
false
true

Variable	Refers to
teamA	object 1 (“Lions”)
favorite	object 1 (same object as teamA)
teamB	object 2 (“Lions”, a separate object)

Variables like teamA and favorite, which refer to the same object, are called aliases.

Watch out: two string literals with the same characters, such as "Lions" and "Lions", may share one object, so == can happen to be true. Never rely on this: compare string contents with equals.

4. Comparing a reference with null

A reference variable can hold null, which means it does not refer to any object. Comparing a reference with null, using == or !=, tells you whether it actually refers to an object 2.6.B.2.

String winner = null;
if (winner == null) {
    System.out.println("No winner yet");
}
winner = "Tigers";
if (winner != null) {
    System.out.println("Winner: " + winner + " (" + winner.length() + " letters)");
}

Output:

No winner yet
Winner: Tigers (6 letters)

Calling a method such as length() on a null reference throws a NullPointerException. Check for null first, and put that check on the left of && so short-circuit evaluation protects the call: winner != null && winner.length() > 2.

5. equals: equivalency by attributes

Classes often define their own equals method to specify what makes two objects of that class equivalent. The decision is most often based on attributes of the two objects 2.6.B.3.

The String class is the example you use most. Its equals method treats two strings as equivalent when they have the same sequence of characters, even if they are separate objects.

String typed = new String("open sesame");
String password = "open sesame";
System.out.println(typed == password);
System.out.println(typed.equals(password));
System.out.println(typed.equals("Open sesame"));

Output:

false
true
false

Uppercase and lowercase letters are different characters, so the last comparison is false.

Other classes choose their own criteria. A class for library cards might call two cards equal when their card numbers match; a class for points on a grid might compare both coordinates. When an exam question gives you a class with an equals method, read its description to see which attributes it compares. (Writing your own replacement for the equals method is outside the AP® course and exam.)

Question you are asking	Use
Do these two variables refer to the same object?	==
Does this variable refer to any object?	== null or != null
Are these two objects equivalent (same attribute values)?	equals