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Placement Test Practice — Set Theory & Logic
This practice test covers set operations, propositional logic truth tables, De Morgan's laws, quantifier negation, and proof techniques. Apply Venn diagrams, construct truth tables, negate quantified statements, and outline proofs using direct, contrapositive, contradiction, and induction methods.
Placement Test Practice — Set Theory & Logic
Practice Test — 20 Questions
1. $A = \{1,2,3\}$, $B = \{2,3,4\}$. $A \cap B = $?
2. $|A \cup B|$ if $|A|=10$, $|B|=7$, $|A \cap B|=3$?
3. Negate: $\forall x\, P(x)$.
4. Truth value: $T \to F$?
5. Is $p \lor \lnot p$ a tautology?
6. $(A \cup B)' = $?
7. Contrapositive of "if it rains, the ground is wet"?
8. $A - B$ when $A \subseteq B$?
9. How many elements in $\mathcal{P}(\{1,2,3\})$?
10. Is $\{\emptyset\} = \emptyset$?
11. $p \land q$ is true only when?
12. Negate: $\exists x\,(x^2 = 4)$.
13. Base case of induction for $n \geq 1$?
14. Direct proof: product of two odd numbers is odd.
15. What is a counterexample?
16. $A \cup A' = $?
17. $p \to q$ is equivalent to?
18. How many rows in a truth table with 4 variables?
19. What does QED or ∎ mean?
20. Negate: "No student failed."
Show Answer Key
1. $\{2,3\}$
2. $14$
3. $\exists x\, \lnot P(x)$
4. False
5. Yes
6. $A' \cap B'$
7. "If the ground is not wet, it did not rain."
8. $\emptyset$
9. $2^3 = 8$
10. No — $\{\emptyset\}$ has one element
11. Both $p$ and $q$ are true
12. $\forall x\,(x^2 \neq 4)$
13. $n = 1$
14. $(2a+1)(2b+1) = 4ab+2a+2b+1 = 2(2ab+a+b)+1$, odd ∎
15. A specific case that disproves a universal statement
16. $U$ (the universal set)
17. $\lnot p \lor q$
18. $16$
19. "Which was to be demonstrated" — marks end of proof
20. "Some student failed."