1
From: "Human Potential & Development."
Split Justification: Development fundamentally involves both our inner landscape (**Internal World**) and our interaction with everything outside us (**External World**). (Ref: Subject-Object Distinction)..
2
From: "Internal World (The Self)"
Split Justification: The Internal World involves both mental processes (**Cognitive Sphere**) and physical experiences (**Somatic Sphere**). (Ref: Mind-Body Distinction)
3
From: "Cognitive Sphere"
Split Justification: Cognition operates via deliberate, logical steps (**Analytical Processing**) and faster, intuitive pattern-matching (**Intuitive/Associative Processing**). (Ref: Dual Process Theory)
4
From: "Analytical Processing"
Split Justification: Analytical thought engages distinct symbolic systems: abstract logic and mathematics (**Quantitative/Logical Reasoning**) versus structured language (**Linguistic/Verbal Reasoning**).
5
From: "Quantitative/Logical Reasoning"
Split Justification: Logical reasoning can be strictly formal following rules of inference (**Deductive Proof**) or drawing general conclusions from specific examples (**Inductive Reasoning Case Study**). (L5 Split)
6
From: "Deductive Proof."
Split Justification: Deductive systems can be analyzed based on the relationship between whole statements (**Propositional Logic**) or the properties of objects and their relations (**Predicate Logic**). (L6 Split)
7
From: "Predicate Logic"
Split Justification: Predicate logic extends reasoning to include variables and quantities (**Understanding Quantifiers**) and applying these to sets of objects (**Basic Set Theory Proof**).
8
From: "Basic Set Theory Proof"
Split Justification: This dichotomy distinguishes between two fundamental methodologies for constructing basic set theory proofs: element-wise proofs, which analyze the membership of individual elements using predicate logic definitions of set operations, and algebraic proofs, which manipulate set expressions using established set identities and laws. These two approaches represent distinct, yet comprehensive, methods for proving set theoretic statements.
9
From: "Element-wise Set Theory Proof"
Split Justification: Element-wise set theory proofs fundamentally aim to establish either that two sets are identical (equality) or that one set is contained within another (inclusion). These represent the two distinct primary objectives of such proofs, which are mutually exclusive in their overall goal and together cover the scope of element-wise set theory proofs.
10
From: "Proof of Set Inclusion"
Split Justification: This fundamental dichotomy distinguishes between proving set inclusion by directly assuming an element belongs to the first set and then logically demonstrating its membership in the second set, versus assuming the negation of set inclusion and deriving a contradiction.
11
From: "Direct Proof of Set Inclusion"
Split Justification: This dichotomy categorizes direct proofs of set inclusion based on the primary logical method used to demonstrate membership in the superset. "Direct Proof by Predicate Deduction" involves showing that an arbitrary element, by virtue of its membership in the subset, directly satisfies the defining predicate or property of the superset through logical or mathematical inference. "Direct Proof by Set Operation Expansion" involves expanding and manipulating the logical conditions derived from set operations (union, intersection, complement, difference) to demonstrate that the arbitrary element satisfies the membership criteria of the superset. These represent distinct modes of deriving the consequence (element is in the superset) from the premise (element is in the subset).
12
From: "Direct Proof by Set Operation Expansion"
Split Justification: This dichotomy classifies the expansion of set operations based on their arity (number of operands), fundamentally distinguishing between operations acting on a single set (unary, e.g., complement) and those acting on two sets (binary, e.g., union, intersection, difference), thereby comprehensively covering all standard set operations relevant to direct proof by expansion.
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Topic: "Expansion of Binary Set Operations" (W7519)