Question

(De Morgan's Laws) In this chapter, we discussed the logical operators 88,11 and \(1 .\) De Morgan's laws can sometimes make it more convenient for us to express a logical expression. These laws state that the expression 1 condition1 ss condition2 ) is logically equivalent to the expression 1 condition1 \(\|\) condition2 ). Also, the expression \(!\) ( condition1 ? I condition? is logically equivalent to the expression ( i condition1 ss ! condition2) . Use De Morgan's laws to write equivalent expressions for each of the following, then write a program to show that the original expression and the new expression in each case are equivalent: a. \(|(x<5)\) ss \(|(y>-7)\) b. \(1(a=b)|| 1(g |=5)\) C. \(1(x \in 8)\) ss \((y>4)\) d. \(1(\quad(i>4), \| \quad(j \in=6))\)

Short answer

Use De Morgan's laws to simplify each expression as stated in the solution steps. Validate equivalence with a test program.

Step-by-step solution

Simplify Expression a Using De Morgan's Law

Expression a is \(\lnot((x < 5) \land (y > -7))\). By De Morgan's Law, \( \lnot(A \land B) \) is equivalent to \((\lnot A) \lor (\lnot B)\). Thus, \(\lnot((x < 5) \land (y > -7))\) is equivalent to \((x \geq 5) \lor (y \leq -7)\).

Simplify Expression b Using De Morgan's Law

Expression b is \(\lnot((a = b) \lor (g eq 5))\). By De Morgan's Law, \( \lnot(A \lor B) \) is equivalent to \((\lnot A) \land (\lnot B)\). Thus, \(\lnot((a = b) \lor (g eq 5))\) is equivalent to \((a eq b) \land (g = 5)\).

Simplify Expression c Using De Morgan's Law

Expression c is \(\lnot((x eq 8) \land (y > 4))\). By De Morgan's Law, \( \lnot(A \land B) \) is equivalent to \((\lnot A) \lor (\lnot B)\). Thus, \(\lnot((x eq 8) \land (y > 4))\) is equivalent to \((x = 8) \lor (y \leq 4)\).

Simplify Expression d Using De Morgan's Law

Expression d is \(\lnot((i > 4) \lor (j eq 6))\). By De Morgan's Law, \( \lnot(A \lor B) \) is equivalent to \((\lnot A) \land (\lnot B)\). Thus, \(\lnot((i > 4) \lor (j eq 6))\) is equivalent to \((i \leq 4) \land (j = 6)\).

Write a Program to Compare Expressions

Write a program in a language like Python to check if the simplified expressions are equivalent to the original ones by testing various input values. ```python # Example in Python x, y, a, b, g, i, j = 3, -8, 5, 5, 6, 3, 6 # Original expressions a_expr = not ((x < 5) and (y > -7)) b_expr = not ((a == b) or (g != 5)) c_expr = not ((x != 8) and (y > 4)) d_expr = not ((i > 4) or (j != 6)) # Simplified expressions a_simp = (x >= 5) or (y <= -7) b_simp = (a != b) and (g == 5) c_simp = (x == 8) or (y <= 4) d_simp = (i <= 4) and (j == 6) # Check equivalence print(a_expr == a_simp) # Should print True print(b_expr == b_simp) # Should print True print(c_expr == c_simp) # Should print True print(d_expr == d_simp) # Should print True ``` This program validates that each original expression is equivalent to its respective simplified expression using De Morgan's laws by checking the logic equivalence with a test case.

Key concepts

Logical Operators

Logical operators in C++ are essential for forming complex boolean expressions. These operators are:

• && (AND): This operator returns true if both conditions are true.
• || (OR): This operator returns true if at least one condition is true.
• ! (NOT): This operator inverts the truth value of a condition.

Using these operators, programmers can create complex expressions to control the flow of their programs. For example, the expression \((x > 5) \land (y < 3)\) returns true only if both \(x > 5\) and \(y < 3\) are true. Logical operators are often used within control structures like if statements and loops to perform logical tests.

Boolean Expressions

Boolean expressions are expressions in programming that evaluate to a Boolean value—either true or false. They are foundational in control flow and decision-making processes of a program.
In C++ and most other programming languages, Boolean expressions combine variables with logical operators to run different blocks of code depending on the result of the expressions. For instance, consider the condition \((a == b) \lor (c != d)\). This evaluates to true if \(a\) is equal to \(b\) or \(c\) is not equal to \(d\).
Boolean expressions often involve comparison operators like

• == (equal to)
• != (not equal to)
• < (less than)
• <= (less than or equal to)
• >> (greater than)
• >= (greater than or equal to)

These expressions are crucial for guiding the logical flow of any application.

Programming Logic

Programming logic refers to the process of using a sequence of instructions to interactively and effectively solve a problem or perform a task. It involves the strategic use of logical operators, operators, and flow control to create programs that make decisions based on given inputs.
Using De Morgan's laws is a common approach in programming logic to simplify complex Boolean expressions. De Morgan's laws state that:

• The negation of a conjunction (AND) is the disjunction (OR) of the negations: \(\lnot(A \land B) = \lnot A \lor \lnot B\)
• The negation of a disjunction (OR) is the conjunction (AND) of the negations: \(\lnot(A \lor B) = \lnot A \land \lnot B\)

These laws allow programmers to refactor expressions for better readability and maintainability in code.
Applying De Morgan's laws helps reveal the logical equivalence between complex expressions, making code easier to understand and debug. By simplifying logical statements, we make programming more efficient and reduce potential errors. These transformations are valuable tools in logical reasoning within program development.