Answer to Question #264578 in Discrete Mathematics for Abhijeet Kaur

Question #264578

For each proposition below, decide whether it is true or false and give a brief explanation. Assume the universe (domain of variables) to be Z, the set of integers.

(1) ((x = 5) ∧ (y = 1)) → ((x > 10) ∨ (y > 0))

(2) ∀x((x < 0) ∨ (x^2 ≥ x))

(3) ¬(∃xP(x)) ↔ (∀x¬P(x)) for all predicates P(x)





1
Expert's answer
2021-11-12T08:22:06-0500

For each proposition below, let us decide whether it is true or false. Assume thedomain of variables to be Z\Z, the set of integers.


(1) ((x=5)(y=1))((x>10)(y>0))((x = 5) ∧ (y = 1)) → ((x > 10) ∨ (y > 0))

The conjunction (x=5)(y=1)(x = 5) ∧ (y = 1) is true if and only if x=5x=5 and y=1.y=1. In this case the disjunction (x>10)(y>0)=(5>10)(1>0)(x > 10) ∨ (y > 0)=(5 > 10) ∨ (1 > 0) is true, and hence the implication ((x=5)(y=1))((x>10)(y>0))((x = 5) ∧ (y = 1)) → ((x > 10) ∨ (y > 0)) is also true. In other cases the conjunction (x=5)(y=1)(x = 5) ∧ (y = 1) is false, and hence the implication ((x=5)(y=1))((x>10)(y>0))((x = 5) ∧ (y = 1)) → ((x > 10) ∨ (y > 0)) is true. Therefore, for all integers xx and yy the statement ((x = 5) ∧ (y = 1)) → ((x > 10) ∨ (y > 0)) is true.

Answer: true


(2) x((x<0)(x2x))∀x((x < 0) ∨ (x^2 ≥ x))

Since for each integer xx the statement x2xx^2\ge x is true, we conclude the disjunction (x<0)(x2x)(x < 0) ∨ (x^2 ≥ x) is also true, and hence the statement x((x<0)(x2x))∀x((x < 0) ∨ (x^2 ≥ x)) is true.

Answer: true


(3) ¬(xP(x))(x¬P(x))¬(∃xP(x)) ↔ (∀x¬P(x)) for all predicates P(x)P(x)

Since ¬(xP(x))(x¬P(x)),¬(∃xP(x)) \equiv (∀x¬P(x)), the statements ¬(xP(x))¬(∃xP(x)) and (x¬P(x))(∀x¬P(x)) have the same values, and hence the equivalence ¬(xP(x))(x¬P(x))¬(∃xP(x)) ↔ (∀x¬P(x)) is true.

Answer: true


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