Problem 13 - Discrete Mathematics with Applications

Question

Let \(x, y,\) and \(z\) be any real numbers. Represent each sentence symbolically, where \(p: xx)]$$

Step-by-Step Solution

Answer

\(\sim p \wedge (q \vee r): (x \geq y) \wedge [(y < z) \vee (x < z)]\)

Step 1: Represent the negation of p symbolically

To get the negation of p, we will use the negation operator (~), denoted as: \[\sim p: x \geq y\]

Step 2: Represent the disjunction of q and r symbolically

To represent the statement "either q or r," we will use the disjunction operator (v), denoted as: \[q \vee r: (y < z) \vee (x < z)\]

Step 3: Combine the negation of p and the disjunction of q and r using "and"

To represent the entire compound sentence, we will combine the negation of p and the disjunction of q and r using the conjunction operator (^), denoted as: \[\sim p \wedge (q \vee r): (x \geq y) \wedge [(y < z) \vee (x < z)]\] So, the given sentence is symbolically represented by \(\sim p \wedge (q \vee r)\).

Key Concepts

Negation

Negation is a fundamental concept in symbolic logic that involves reversing the truth value of a statement. If a statement is true, its negation is false, and vice versa. This is essential in logic as it allows for the construction of complex statements by altering simple propositions.

When dealing with a statement like "p: \(x < y\)", its negation is represented by adding a tilde (~) symbol in front of the statement, giving us "\(\sim p\)". In this case, if "p" means "x is less than y", then "\(\sim p\)" means "x is not less than y", which can be written as "x is greater than or equal to y" or "\(x \geq y\)". This transformation flips the meaning of the statement.

Negations are important as they allow us to express situations where specific conditions are not met, facilitating a more comprehensive understanding of possible outcomes.

Disjunction

Disjunction is a logical operation that connects two statements with the word "or." In symbolic logic, this is represented using the operator \(\vee\), which denotes an inclusive "or". This means that at least one of the statements must be true for the entire disjunction to be considered true.

For example, given the statements "q: \(y < z\)" and "r: \(x < z\)", the disjunction of "q" and "r" is represented as "\(q \vee r\)" or more explicitly, "\((y < z) \vee (x < z)\)". Therefore, the truth of the disjunction depends on either "y is less than z" or "x is less than z" being true—or both.

This concept is particularly useful in decision-making and conditional statements where multiple pathways or options are available.

Conjunction

Conjunction in symbolic logic refers to the combination of two statements into one comprehensive statement using the word "and." It is represented by the operator \(\wedge\), requiring both individual statements to be true for the whole conjunction to be true.

Let's consider the negation of "p" and the disjunction of "q" and "r". We combine them using a conjunction. The expression is depicted symbolically as "\(\sim p \wedge (q \vee r)\)". Here, "\(\sim p\)" is "\(x \geq y\)", and "\(q \vee r\)" is the previously discussed disjunction "\((y < z) \vee (x < z)\)".

This combination signifies that both conditions need to hold: "x is greater than or equal to y," and at least one condition between "y is less than z" and "x is less than z" is true. Conjunctions allow us to form comprehensive logical statements that more closely mirror complex real-world conditions.