The set D1 consists of three students: Garth, who is 5 feet 11 inches tall; Erin, who is 5 feet 6 inches tall; and Marts, who is 6 feet tall. The set D2 consists of four students: Dale, who is 6 feet tall; Garth, who is 5 feet II inches tall; Erin, who is 5 feet 6 inches tall; and Marty, who is 6 feet tall. The set D3 consists of one student: Dale, who is 6 feet tall. The set D4 consists of three students: Pat, Sandy, and Gale, each of whom is 5 feet 11 inches tall.
T3(x, y) is the propositional function "if x and y are distinct persons, then x is taller than y.”
Tell whether each proposition in Exercises 1–4 is true or false if the domain of discourse is D3 × D3. (The set D3 is defined before Exercise 5.)
Exercise 1
The set D1 consists of three students: Garth, who is 5 feet 11 inches tall; Erin, who is 5 feet 6 inches tall; and Marts, who is 6 feet tall. The set D2 consists of four students: Dale, who is 6 feet tall; Garth, who is 5 feet II inches tall; Erin, who is 5 feet 6 inches tall; and Marty, who is 6 feet tall. The set D3 consists of one student: Dale, who is 6 feet tall. The set D4 consists of three students: Pat, Sandy, and Gale, each of whom is 5 feet 11 inches tall.
T3(x, y) is the propositional function "if x and y are distinct persons, then x is taller than y.” Write each proposition in words.
∀x∀yT3(x, y)
Exercise 2
The set D1 consists of three students: Garth, who is 5 feet 11 inches tall; Erin, who is 5 feet 6 inches tall; and Marts, who is 6 feet tall. The set D2 consists of four students: Dale, who is 6 feet tall; Garth, who is 5 feet II inches tall; Erin, who is 5 feet 6 inches tall; and Marty, who is 6 feet tall. The set D3 consists of one student: Dale, who is 6 feet tall. The set D4 consists of three students: Pat, Sandy, and Gale, each of whom is 5 feet 11 inches tall.
T3(x, y) is the propositional function "if x and y are distinct persons, then x is taller than y.” Write each proposition in words.
∀xƎyT3(x, y)
Exercise 3
The set D1 consists of three students: Garth, who is 5 feet 11 inches tall; Erin, who is 5 feet 6 inches tall; and Marts, who is 6 feet tall. The set D2 consists of four students: Dale, who is 6 feet tall; Garth, who is 5 feet II inches tall; Erin, who is 5 feet 6 inches tall; and Marty, who is 6 feet tall. The set D3 consists of one student: Dale, who is 6 feet tall. The set D4 consists of three students: Pat, Sandy, and Gale, each of whom is 5 feet 11 inches tall.
T3(x, y) is the propositional function "if x and y are distinct persons, then x is taller than y.” Write each proposition in words.
Ǝx∀yT3(x, y)
Exercise 4
The set D1 consists of three students: Garth, who is 5 feet 11 inches tall; Erin, who is 5 feet 6 inches tall; and Marts, who is 6 feet tall. The set D2 consists of four students: Dale, who is 6 feet tall; Garth, who is 5 feet II inches tall; Erin, who is 5 feet 6 inches tall; and Marty, who is 6 feet tall. The set D3 consists of one student: Dale, who is 6 feet tall. The set D4 consists of three students: Pat, Sandy, and Gale, each of whom is 5 feet 11 inches tall.
T3(x, y) is the propositional function "if x and y are distinct persons, then x is taller than y.” Write each proposition in words.
ƎxƎyT3(x, y)
Exercise 5
The set D1 consists of three students: Garth, who is 5 feet 11 inches tall; Erin, who is 5 feet 6 inches tall; and Marts, who is 6 feet tall. The set D2 consists of four students: Dale, who is 6 feet tall; Garth, who is 5 feet II inches tall; Erin, who is 5 feet 6 inches tall; and Marty, who is 6 feet tall. The set D3 consists of one student: Dale, who is 6 feet tall. The set D4 consists of three students: Pat, Sandy, and Gale, each of whom is 5 feet 11 inches tall.
T1(x, y)is the propositional function “x is taller than y.” Write each proposition in words.
∀x∀y T1(x, y)
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