In this chapter, we discussed the logical operators \(\& \&, \&,||, |, \wedge\) and \(!\) De Morgan's laws can sometimes make it more convenient for us to express a logical expression. These laws state that the expression ! ( condition 1 \&\& condition2) is logically equivalent to the expression \((! \text { condition } 1 \text { I } | \text { ! condition } 2 \text { ). Also, the expression } ! \text { (condition1 }\) | | condition2) is logically equivalent to the expression ( 1 condition 1 88 ! condition2). Use De Morgan's laws to write equivalent expressions for each of the following, then write an application to show that both the original expression and the new expression in each case produce the same value: a) !(x < 5) && !(y >= 7) b) !(a == b) || !(g != 5) c) !((x <= 8) && (y > 4)) d) !((i > 4) || (j <= 6))

Step by step solution

01

Understand De Morgan's Laws

De Morgan's laws can make it easier to express logical expressions. The laws are: \(eg (p \land q) \equiv (eg p) \lor (eg q)\) and \(eg (p \lor q) \equiv (eg p) \land (eg q)\).

02

Apply De Morgan's Law to expression a)

Given expression: \(!(x < 5) \land !(y \geq 7)\). Apply De Morgan's Law to get the equivalent expression: \((x \geq 5) \lor (y < 7)\).

03

Apply De Morgan's Law to expression b)

Given expression: \(!(a == b) \lor !(g eq 5)\). Apply De Morgan's Law to get the equivalent expression: \((a eq b) \land (g == 5)\).

04

Apply De Morgan's Law to expression c)

Given expression: \(!((x \leq 8) \land (y > 4))\). Apply De Morgan's Law to get the equivalent expression: \((x > 8) \lor (y \leq 4)\).

05

Apply De Morgan's Law to expression d)

Given expression: \(!((i > 4) \lor (j \leq 6))\). Apply De Morgan's Law to get the equivalent expression: \((i \leq 4) \land (j > 6)\).

06

Write Application to Test Equivalence

To show that the original expressions and the new expressions produce the same values, write a program that tests each pair of equivalent expressions for different sets of variables to confirm that their results are identical.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Logical Operators in Java

In Java programming, logical operators are used to connect boolean expressions and return another boolean value. They play a critical role in making decisions within the code. There are several logical operators, including && (logical AND), || (logical OR), ! (logical NOT), and ^ (logical XOR).
With &&, both operands must be true for the result to be true, while with ||, only one operand needs to be true. The ! operator inverts the truth value of its operand, and ^ evaluates to true if the operands are distinct; otherwise, it's false.
Understanding these operators helps in creating efficient boolean expressions and is a stepping stone to mastering De Morgan's Laws, which provide rules for simplifying complex logical expressions.

Boolean Expressions

Boolean expressions in Java are statements that evaluate to either true or false. These are commonly found in control flow statements and are used for making decisions in the code. To construct a boolean expression, relational operators such as <, >, ==, or != are often paired with logical operators.
Understanding boolean expressions is essential, as they form the backbone of decision-making processes within Java programs. For instance, an expression (x < 5) evaluates to true if x is less than 5. When combined with logical operators and possibly negated, boolean expressions become more sophisticated, which leads to the application of De Morgan's Laws for simplification as seen in the provided exercise solutions.

Java Control Statements

Control statements in Java govern the flow of execution of the program. They can be classified into several types, including conditional statements like if, else, and switch; loops such as for, while, and do-while; and control transfer statements like break, continue, and return.
Using these control statements effectively requires a clear grasp of boolean expressions and logical operators. For example, an if statement might use a boolean expression that relies on De Morgan's Law to determine which block of code to execute. Understanding how to rewrite complex logical expressions using De Morgan's can make your code more readable and maintainable—a vital skill for Java programmers.