7. Punctuation: the comma, and (right and left) parentheses ),( As predicate is the most important one, it is called predicate logic. It is named first order logic embedded into predicate logic. In another word, proposition logic is a subset of predicate logic To formalize the examples shown to introduce predicate logic, we define the following definitions Definition 2(Term). Terms 1. Every variable is a term 2. Every constants symbol is a term 3. If f is an n-ary function symbol (n= 1, 2,...) and t1,.. tn are terms, then f(t1, ., tn) s also a term Definition 3(Ground term). Terms with no variables are called variable-free terms or ground Definition 4(Atomic formula). An atomic formula is an expression of the form R(t1, ...,tn) where R is an n-ary predicate symbol and ti, .. tn are terms Definition 5 (Formula). Formalas 1. Every atomic formula is a formula 2. If a, B are formulas, then so are(oA B),(a>),a+B), a), (avB) 3. If v is a variable and a is a formula, then(vja)and((vu)a) are also formulas With predicate logic, many mathematical statement can be formalized. Consider the followin Example 5. Let the domain consist of all relational numbers @. Again p (a, y)=(a<y), f(a, y) +y, g(a, y)=a:y and a=0, b= l, c=2 are constants 1.(y(x,y)∧y(y,2) 2.(y)(y(x,y)∧y(y,2) (vx)(y(x,2)→(y)(yp(x,y)∧φ(y,2) 4.(x)(wy)(y(x,y)→(y(x,y(f(x,y),c)∧φ(g(f(x,y),c),y) 5.9(3,f(3,y)7. Punctuation: the comma , and (right and left) parentheses ), ( As predicate is the most important one, it is called predicate logic. It is named first order logic together with proposition logic. In the latter, we will show you that proposition logic can be embedded into predicate logic. In another word, proposition logic is a subset of predicate logic. To formalize the examples shown to introduce predicate logic, we define the following definitions. Definition 2 (Term). Terms. 1. Every variable is a term 2. Every constants symbol is a term. 3. If f is an n-ary function symbol ( n = 1, 2, . . . ) and t1, . . . , tn are terms, then f(t1, . . . , tn) is also a term. Definition 3 (Ground term). Terms with no variables are called variable-free terms or ground terms. Definition 4 (Atomic formula). An atomic formula is an expression of the form R(t1, . . . , tn) where R is an n-ary predicate symbol and t1, . . . , tn are terms. Definition 5 (Formula). Formulas. 1. Every atomic formula is a formula. 2. If α, β are formulas, then so are (α ∧ β),(α → β),(α ↔ β),(¬α),(α ∨ β). 3. If v is a variable and α is a formula, then ((∃v)α) and ((∀v)α) are also formulas. With predicate logic, many mathematical statement can be formalized. Consider the following example. Example 5. Let the domain consist of all relational numbers Q. Again ϕ(x, y) = (x < y), f(x, y) = x + y, g(x, y) = x ÷ y and a = 0, b = 1, c = 2 are constants. 1. (ϕ(x, y) ∧ ϕ(y, z)) 2. ((∃y)(ϕ(x, y) ∧ ϕ(y, z))) 3. ((∀x)(ϕ(x, z) → ((∃y)(ϕ(x, y) ∧ ϕ(y, z)))) 4. ((∀x)((∀y)(ϕ(x, y) → (ϕ(x, g(f(x, y), c)) ∧ ϕ(g(f(x, y), c), y))))) 5. ϕ(y, f(y, y)) 4