## Mathematical Expressions and Theorem's Conclusion
### Overview
The image presents a table comparing mathematical expressions for logical operations (AND, OR, NAND, NOR, XOR) with their corresponding theorem conclusions. The left column displays the mathematical formulas, while the right column shows the theorem's conclusion related to each logical operation.
### Components/Axes
The table has two columns:
* **Mathematical Expressions**: Contains mathematical formulas for FAND, FOR, FNAND, FNOR, and FXOR.
* **Theorem's Conclusion**: Contains logical expressions and functions related to the mathematical expressions.
### Detailed Analysis
**Mathematical Expressions Column:**
* **FAND(t)**:
* `FAND(t) = Pr(∩(i=2 to N) Ai(t))`
* `= Π(i=2 to N) Fi(t)`
* **FOR(t)**:
* `FOR(t) = Pr(∪(i=2 to N) Ai(t))`
* `= 1 - Π(i=2 to N) (1 - Fi(t))`
* **FNAND(t)**:
* `FNAND(t) = Pr(∩(i=2 to k) Āi(t) ∩ ∩(j=k to N) Aj(t))`
* `= Π(i=2 to k) (1 - Fi(t)) * Π(j=k to N) (Fj(t))`
* **FNOR(t)**:
* `FNOR(t) = 1 - FOR(t) = Π(i=2 to N) (1 - Fi(t))`
* **FXOR(t)**:
* `FXOR(t) = Pr(Ā(t)B(t) ∪ A(t)B̄(t))`
* `= (1 - FA(t))FB(t) + FA(t)(1 - FB(t))`
**Theorem's Conclusion Column:**
* **AND**:
* `∀ p L1 L2.`
* `(prob p (AND_FT_gate L) = list_prod (list_prob p L))`
* **OR**:
* `∀ p L1 L2.`
* `(prob p (OR_FT_gate p L) = 1 - list_prod (one_minus_list (list_prob p L)))`
* **NAND**:
* `∀ p L1 L2.`
* `(prob p (NAND_FT_gate p L1 L2) = list_prod (list_prob p (compl_list p L1)) * list_prod (list_prob p L2))`
* **NOR**:
* `∀ p L.`
* `(prob p (NOR_FT_gate p L) = list_prod (one_minus_list (list_prob p L)))`
* **XOR**:
* `∀ p A B.`
* `prob_space p ∧ A ∈ events p ∧ B ∈ events p ⇒ (prob p (XOR_FT_gate p A B) = (1 - prob p A) * prob p B + prob p A * (1 - prob p B))`
### Key Observations
* The mathematical expressions define the probability of AND, OR, NAND, NOR, and XOR operations in terms of events A and B.
* The theorem's conclusions provide logical representations of these operations, likely within a formal system or programming context.
* The use of `list_prod`, `list_prob`, and `one_minus_list` suggests these are functions operating on lists of probabilities.
* The NAND gate conclusion uses `compl_list`, suggesting a function that complements a list of probabilities.
### Interpretation
The table bridges the gap between probability theory and logical operations, providing a formal representation of how these operations can be expressed and reasoned about within a probabilistic framework. The theorem's conclusions likely serve as a basis for implementing these logical operations in systems that deal with probabilities, such as fault-tolerant systems or probabilistic programming languages. The use of list operations suggests that these operations can be generalized to handle multiple inputs.
