10.16 Propositions and Quantifiers

Exercise 1

Write the denial of the following statement "Even the computer is not working or the headphones are not on"

Let \(P=\)"The computer is working", \(Q=\)"The headphones are on"

\(\neg P\vee \neg Q\)

The denial is \(\neg(\neg P\vee \neg Q)\Rightarrow P\wedge Q\)

The denial is "The computer is working and the headphones are on"

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Exercise 2

Write the converse and contrapositive of "If \(1+1=2\Rightarrow \sqrt{10}>3\)\"

\(P=\)"\(1+1=2\)", \(Q=\)"\(\sqrt{10}>3\)\"

\(P\Rightarrow Q\)

Converse: \(Q\Rightarrow P\), then it is \(\sqrt{10}>3\Rightarrow1+1=2\)​

Contrapositive: \(\neg Q\Rightarrow \neg P\), then it is \(\sqrt{10}\leq3\Rightarrow1+1\neq2\)

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Exercise 3

Universe: \(\N\), open propositions: \(A(x)="x>3",B(x)="x\geq 4"\)

1. Are \(A(x)\) and \(B(x)\) equivalent open sentences

   The truth set of \(A(x)\) is \(\{4,5,6,...\}=\N_{\geq 4}\)​

   Yes, they are equivalent because the truth set is \(\N_{\geq 4}\) for both open propositions

2. \(\forall x, A(x)\) \(\forall x, B(x)\)

   Are these two quantified sentences equivalent?

   \(\forall x, A(x)\) False

   \(\forall x, B(x)\) False

   Yes, they are equivalent because they are both False

3. Write the denial in English and with logical propositions and say which one is true or false

   \(\forall x (A(x)\vee \neg B(x))\)​

   For any \(n\in \N\)\, either \(n>3\) or \(n>4\) T

   There exists \(n\in \N\) such that \(n\leq 3\) and \(n\geq 4\) F

   \(\exists x(\neg A(x)\wedge B(x))\)

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Exercise 4

1. Translate using quantifiers where the universe is fruits

   \(A(x)=\)"\(x\) is an apple"

   \(B(x)=\)"\(x\) has seeds"

   1. "All apples have seeds"

      \(\forall x (A(x)\Rightarrow B(x))\) 2. "Some apples have seeds"

      \(\exists x(A(x)\wedge B(x))\)

2. Deny \(1\) and \(2\)​

   1. Some apples doesn't have seeds

      \(\exists x(A(x)\wedge \neg B(x))\)​ 2. \(\forall x(\neg A(x)\vee \neg B(x))\)

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