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 Equality"
Split Justification: The fundamental definition of set equality (A = B) is that A is a subset of B (A ⊆ B) AND B is a subset of A (B ⊆ A). Therefore, a proof of set equality naturally decomposes into two distinct and necessary sub-proofs: demonstrating that the first set is a subset of the second, and vice versa. These two tasks are mutually exclusive in their conclusion but together comprehensively cover the parent concept.
11
From: "Proof of Second Set Being a Subset of the First"
Split Justification: This dichotomy separates the two fundamental logical approaches to proving an implication (x ∈ A → x ∈ B). The direct method assumes the premise (x ∈ A) and directly derives the conclusion (x ∈ B). The indirect method relies on strategies such as proof by contradiction (assuming the negation of the conclusion or the entire statement leads to a logical inconsistency) or proof by contrapositive (proving the logically equivalent statement ¬(x ∈ B) → ¬(x ∈ A)). These methods are mutually exclusive in their overall argumentative structure and comprehensively cover all valid element-wise proof techniques for demonstrating set subset relationships.
12
From: "Proof by Indirect Method"
Split Justification: These represent the two primary and distinct logical strategies for constructing an indirect proof. Proof by contradiction assumes the negation of the desired conclusion and derives an inconsistency, whereas proof by contrapositive establishes the logical equivalence of a conditional statement to its contrapositive.
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Topic: "Proof by Contrapositive" (W7775)