Axiom: "Two points can be joined by a straight line."
Axiom: "A finite straight line can be extended indefinitely in a straight line."
Axiom: "A circle can be drawn with any center and radius."
Theorem: "In a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides."
2. Peano's axioms and the commutative property of addition:
Axiom: "Zero is a natural number."
Axiom: "Every natural number has a successor, which is also a natural number."
Axiom: "Zero is not the successor of any natural number."
Theorem: "For any two natural numbers a and b, a + b = b + a."
3. Boolean algebra axioms and De Morgan's theorem:
Axiom: "There are two Boolean values, true and false."
Axiom: "Every Boolean operation is defined by a truth table."
Axiom: "There is a unique Boolean identity element for each operation."
Theorem: "The negation of the conjunction of two propositions is logically equivalent to the disjunction of the negations of the propositions." (i.e., ¬(p ∧ q) ≡ ¬p ∨ ¬q) and vice versa.
Answers & Comments
Answer:
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Step-by-step explanation:
Here are three examples of axioms and theorems:
1. Euclid's axioms and the Pythagorean theorem:
2. Peano's axioms and the commutative property of addition:
3. Boolean algebra axioms and De Morgan's theorem:
Answer:
1)2+2=4, 3 × 3=4
2)2 + 2 = 4, 3 × 3 = 9
3)if x + y = 1 and y + 1 = 9, then x = 9
Step-by-step explanation:
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