1.5 Quantifiers & Predicate Logic

Determine which propositions are true when the variables range over:

1. the nonnegative integers.

   Exercise 1

     \(\forall x \exists y. 2x-y = 0\)  

     \(\forall x \exists y. x-2y = 0\)  

   &nbsp; \(\forall x. x < 10 \text{ IMPLIES } [\forall y. y < x \text{ IMPLIES } y < 9]\) &nbsp;

   &nbsp; \(\forall x \exists y. [y >x \text{ AND } \exists z. y+z = 100]\) &nbsp;

   Counterexample for 2: Let \(x = 1\).

   Counterexample for 4: Let \(x = 100\).

2. the integers.

   Exercise 2

   &nbsp; \(\forall x \exists y. 2x-y = 0\) &nbsp;

   &nbsp; \(\forall x \exists y. x-2y = 0\) &nbsp;

   &nbsp; \(\forall x. x < 10 \text{ IMPLIES } [\forall y. y < x \text{ IMPLIES } y < 9]\) &nbsp;

   &nbsp; \(\forall x \exists y. [y >x \text{ AND } \exists z. y+z = 100]\) &nbsp;

   Counterexample for 2: Let \(x = 1\).

3. the real numbers.

   Exercise 3

   &nbsp; \(\forall x \exists y. 2x-y = 0\) &nbsp;

   &nbsp; \(\forall x \exists y. x-2y = 0\) &nbsp;

   &nbsp; \(\forall x. x < 10 \text{ IMPLIES } [\forall y. y < x \text{ IMPLIES } y < 9]\) &nbsp;

   &nbsp; \(\forall x \exists y. [y >x \text{ AND } \exists z. y+z = 100]\) &nbsp;

   Counterexample for 3: Let \(x = 9.5\).

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