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Logic

Evaluating conjunctions, disjunctions and negations of concrete true-or-false propositions, building toward implications and quantified predicate claims.

A Cool Brisk Walk Through Discrete Mathematics, Logic

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36 exercises

Exercise 1

Let BB be the proposition that Joe Biden was elected president in 2020, CC be the proposition that Covid-19 was completely and permanently eradicated from the earth in 2021, and RR be the proposition that Roe v. Wade was overturned in 2022.

What's B∨CB \vee C?

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Exercise 2

What's B∧CB \wedge C?

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Exercise 3

What's B∧RB \wedge R?

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Exercise 4

What's B∧¬RB \wedge \neg R?

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Exercise 5

What's ¬C∨¬R\neg C \vee \neg R?

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Exercise 6

What's ¬(C∨¬R)\neg (C \vee \neg R)?

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Exercise 7

What's ¬(¬C∨R)\neg (\neg C \vee R)?

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Exercise 8

What's ¬C∨B\neg C \vee B?

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Exercise 9

What's ¬C⊕B\neg C \oplus B?

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Exercise 10

What's ¬C⊕¬B\neg C \oplus \neg B?

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Exercise 11

What's ¬¬¬¬B\neg \neg \neg \neg B?

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Exercise 12

What's ¬¬¬¬¬B\neg \neg \neg \neg \neg B?

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Exercise 13

What's ¬¬¬¬¬C\neg \neg \neg \neg \neg C?

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Exercise 14

What's B∨C∨RB \vee C \vee R?

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Exercise 15

What's B∧C∧RB \wedge C \wedge R?

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Exercise 16

What's B∧¬C∧RB \wedge \neg C \wedge R?

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Exercise 17

What's B⇒RB \Rightarrow R?

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Exercise 18

What's R⇒BR \Rightarrow B?

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Exercise 19

What's B⇒CB \Rightarrow C?

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Exercise 20

What's C⇒BC \Rightarrow B?

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Exercise 21

What's C⇒¬RC \Rightarrow \neg R?

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Exercise 22

What's C⇔BC \Leftrightarrow B?

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Exercise 23

What's C⇔¬BC \Leftrightarrow \neg B?

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Exercise 24

I make this assertion:

“X∧¬Y∧¬(Z⇒Q)X \wedge \neg Y \wedge \neg(Z \Rightarrow Q).”

And since I'm the professor, you can assume I'm correct about this. From this information alone, can you determine a unique set of values for the four variables? Or is there more than one possibility for them?

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Exercise 25

What if I get rid of QQ and replace it with XX, thus making my assertion:

“X∧¬Y∧¬(Z⇒X)X \wedge \neg Y \wedge \neg(Z \Rightarrow X).”

Now what is/are the solutions?

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Exercise 26

True or false: ∀x Professor(x)\forall x\ \text{Professor}(x).

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Exercise 27

True or false: ∀x Human(x)\forall x\ \text{Human}(x).

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Exercise 28

True or false: ¬∀x Human(x)\neg \forall x\ \text{Human}(x).

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Exercise 29

True or false: ∀x ¬Human(x)\forall x\ \neg \text{Human}(x).

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Exercise 30

True or false: ∃x ¬Human(x)\exists x\ \neg \text{Human}(x).

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Exercise 31

True or false: ¬∃x Human(x)\neg \exists x\ \text{Human}(x).

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Exercise 32

True or false: ∀x Human(x)∧Professor(x)\forall x\ \text{Human}(x) \wedge \text{Professor}(x).

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Exercise 33

True or false: ∀x Human(x)⇒Professor(x)\forall x\ \text{Human}(x) \Rightarrow \text{Professor}(x).

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Exercise 34

True or false: ∃x Professor(x)⇒Human(x)\exists x\ \text{Professor}(x) \Rightarrow \text{Human}(x).

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Exercise 35

True or false: ∀x Professor(x)⇒Human(x)\forall x\ \text{Professor}(x) \Rightarrow \text{Human}(x).

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Exercise 36

True or false: ¬∃x Professor(x)⇒¬Human(x)\neg \exists x\ \text{Professor}(x) \Rightarrow \neg \text{Human}(x).

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Adapted from A Cool Brisk Walk Through Discrete Mathematics © 2023 Stephen Davies. Licensed under CC BY-SA 4.0. Source: A Cool Brisk Walk Through Discrete Mathematics. Imported at revision 4b9934b.