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#BooleanVariables

Jon AwbreyInquiry@mathstodon.xyz
2022-11-28

#DifferentialPropositionalCalculus • 7.2
• inquiryintoinquiry.com/2020/03

In a #UniverseOfDiscourse based on 3 #BooleanVariables \(p,q,r\) there are \(2^3=8\) #SingularPropositions. Their #VennDiagrams are shown in Figure 10.

\(\text{Figure 10. Singular Propositions} : \mathbb{B}^3 \to\mathbb{B}\)
• inquiryintoinquiry.files.wordp

Related Subjects —
#Logic #LogicalGraphs #DifferentialLogic
#PropositionalCalculus #BooleanFunctions
#CactusSyntax #MinimalNegationOperators
#DiscreteDynamics #QualitativeDynamics

The Figure shows the 8 venn diagrams corresponding to the 8 singular propositions on 3 variables p, q, r. There is 1 diagram at the top of the Figure, 3 in the 2nd row, 3 in the 3rd row, and 1 diagram at the bottom. These represent the singular propositions of rank 3, 2, 1, and 0, respectively. Each of the 8 diagrams consists of a rectangular area representing the universe of discourse and 3 overlapping circular areas P, Q, R representing 3 subsets of that universe. The circular areas are labeled p, q, r respectively to indicate the subsets P, Q, R where the corresponding boolean variables equal 1. The singular proposition of rank 3 corresponds to the boolean product or logical conjunction pqr. The function equals 1 in the intersection of the 3 circular areas and its venn diagram indicates that by coloring the corresponding cell a shade of blue, leaving the remaining area white. The 3 singular propositionss of rank 2 are expressed, respectively, by the following three logical expressions: 1. p and r and not q. 2. q and r and not p. 3. p and q and not r. The 3 singular propositions of rank 1 are expressed, respectively, by the following three logical expressions: 1. q and not p and not r. 2. p and not q and not r. 3. r and not p and not q. The singular proposition of rank 0 is expressed by the logical expression “not p and not q and not r”. The corresponding venn diagram shades the region outside all 3 labeled circular areas, leaving the union of those areas white.
Jon AwbreyInquiry@mathstodon.xyz
2022-11-25

#DifferentialPropositionalCalculus • 6.1
• inquiryintoinquiry.com/2020/03

In a #UniverseOfDiscourse based on 3 #BooleanVariables \(p, q, r\) there are \(2^3 = 8\) #PositivePropositions. Their #VennDiagrams are shown in Figure 9.

\(\text{Figure 9. Positive Propositions} : \mathbb{B}^3 \to \mathbb{B}\)
• inquiryintoinquiry.files.wordp

#Logic #LogicalGraphs #DifferentialLogic
#PropositionalCalculus #BooleanFunctions
#CactusSyntax #MinimalNegationOperators
#DiscreteDynamics #QualitativeDynamics

The Figure shows the 8 venn diagrams corresponding to the 8 positive propositions or positive boolean functions on 3 variables p, q, r. There is 1 venn diagram at the top of the Figure, 3 in the 2nd row, 3 in the 3rd row, and 1 venn diagram at the bottom. These represent the positive propositions of rank 3, 2, 1, and 0, respectively. Each of the 8 venn diagrams consists of a rectangular area representing the universe of discourse and 3 overlapping circular areas P, Q, R representing 3 subsets of that universe. The circular areas are labeled p, q, r respectively to indicate the subsets P, Q, R where the corresponding boolean variables are equal to 1, in other words, true. The positive proposition of rank 3 corresponds to the boolean product or logical conjunction pqr. The function equals 1 or the proposition is true in the intersection of the 3 circular areas & the venn diagram indicates that by coloring that cell a shade of blue, leaving the remaining area white. The 3 positive propositions of rank 2 correspond to the 3 boolean products, pr, qr, pq, respectively. Their diagrams shade the intersections P∩R, Q∩R, P∩Q, respectively. The 3 positive propositions of rank 1 are just the 3 basic propositions, p, q, r and so their diagrams shade the circular areas P, Q, R, respectively. The positive proposition of rank 0 is just the everywhere true proposition or the constant 1 function, so its diagram shades the whole rectangular area representing the universe of discourse.
Jon AwbreyInquiry@mathstodon.xyz
2022-11-21

#DifferentialPropositionalCalculus • 5.2
• inquiryintoinquiry.com/2020/02

In a #UniverseOfDiscourse based on three #BooleanVariables \(p, q, r\) the #LinearPropositions take the shapes of the #VennDiagrams shown in Figure 8. Equivalent verbal & variant logical expressions are given in the next few posts.

\(\text{Figure 8. Linear Propositions} : \mathbb{B}^3 \to \mathbb{B}\)
• inquiryintoinquiry.files.wordp

Related Subjects —
#Logic #LogicalGraphs #DifferentialLogic
#PropositionalCalculus #BooleanFunctions

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