Category Archives: Study Notes

Set Identities: 1 – Commutativity Changing the order of elements won’t affect the results. Example in Arithmetics: 5 + 4 = 4 + 5 = 9 and 3 x 1 = 1 x 3 = 3 The addition and multiplication … Continue reading →

We can use logical equivalences to reduce complex formulas into simpler ones. Two new symbols Identity Law P and True will be true when both P and Tautology are true. P or False will be true when either is true, … Continue reading →

Proofs using Truth Tables Formulas p and q are logically equivalent iff the truth conditions of p are the same as the truth conditions of q. p q x x y y Examples: p q p∧q p∨q ¬(p∨q) 1 1 … Continue reading →

Review: Each statement is TRUE (1) or FALSE (0). All connectives take a truth value and output a truth value.It means that depending on the connective, the truth value can change. Negation The negation is always the opposite of the … Continue reading →

Propositional Logic A statement is a declarative sentence that can be True (1) or False (0). Examples: We cannot express things like questions, as questions cannot be true or false. We also cannot express with imperatives or commands, as those … Continue reading →

De Morgan’s laws describe how mathematical statements and concepts are related through their opposites. In set theory, De Morgan’s laws relate to the intersection and union of sets through their complements. The structure of De Morgan’s laws, whether applied to … Continue reading →

Universal Set A universal set is a set that contains everything. We note the universal set with the letter U. Venn Diagrams A Venn diagram is used to visualize the possible relations among a collection of sets. U is called … Continue reading →

A string is a finite sequence of symbols from an alphabet. A set of binary. The notation above means strings of combinations of 0 and 1. Example: A set of English alphabet. Strings: No limit, sets of strings from the … Continue reading →