Training Discrete Mathematics & Number Theory Logic and Sets
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Logic and Sets

23 min Discrete Mathematics & Number Theory
Logic and set theory form the bedrock of discrete mathematics. Propositions—statements that are true or false—combine through logical connectives (AND, OR, NOT, IMPLIES) to build complex assertions whose truth values can be systematically determined via truth tables. Sets collect distinct objects, and the operations of union, intersection, difference, and complement manipulate these collections. Together, logic and sets provide the formal language for defining mathematical structures, writing precise proofs, and designing digital circuits and database queries.

Logic and Sets

Discrete mathematics starts with statements, truth values, and collections of objects called sets.

Logical Connectives
  • $p \land q$: and
  • $p \lor q$: or
  • $\neg p$: not
  • $p \to q$: if $p$ then $q$
Sets

Common operations are union $A \cup B$, intersection $A \cap B$, and complement $A^c$.

Example 1

If $A=\{1,2,3\}$ and $B=\{3,4,5\}$, find $A \cap B$.

  1. $\{3\}$.
Example 2

What is $A \cup B$ for the same sets?

  1. $\{1,2,3,4,5\}$.

Practice Problems

1. What does $\neg p$ mean?
2. If $A=\{1,2\}$ and $B=\{2,3\}$, find $A \cap B$.
3. Find $A \cup B$ for the same sets.
4. What does $p \land q$ mean?
5. What does complement mean?
Show Answer Key

1. Not $p$

2. $\{2\}$

3. $\{1,2,3\}$

4. Both $p$ and $q$ are true

5. Everything not in the set, relative to the universal set

🔌 Logic Gate Simulator
Output
Expression
Truth table column
Counting and Combinatorics