Discrete Mathematics Foundations: Sets, Logic, Graphs, and Proofs
Speaking Mathematically: Core Concepts
The Language of Sets
- ∀ - Universal quantifier (true for all values of a variable in a set)
- ∃ - Existential quantifier (true for at least one value of a variable in a set)
Common Set Notations:
- R - Set of all real numbers
- Z - Set of all integers
- Q - Set of all rational numbers
- N - Set of all positive integers
- (X)+- - Positive/negative elements for a specific set X
x ∈ S means that x is an element of the set S.
- Example: x=5, S={1, 2, 3, 4, 5}
A ⊆ B means that A is a subset of the set B.
- Example: A={a,b}, B={a,b,c}
- Example: {2} ∈ {1, 2, 3} is false; {2} ∈ {{1}, {2}} is true; 2 ⊆ {1, 2, 3} is false; {2} ⊆ {1, 2, 3} is true
- Note: {9, 9, 1, 1, 7, 7} has only 3 distinct elements ({1, 7, 9}); {1, {2}} has 2 elements;
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