a) The proposition is [tex]\alpha = \left\{ \forall \, x, y \in \mathbb{N} \land \forall \,x, y < 0\,|\,\,xyP, P = \{(x,y)\,|\,x\cdot y > 0\}\right\}[/tex]
b) The proposition is [tex]\beta = \left\{ \forall \, x, y \in \mathbb{N} \land \forall \,x, y > 0\,|\,\,xyP, P = \{(x,y)\,|\,x\cdot y > 0\}\right\}[/tex]
c) The proposition is [tex]\gamma = \left\{\exists \, x, y \in \mathbb{N}\,\land\,\exists \,x,y < 0\,|\,xyP, P = \{(x,y)\,|\, x-y > 0\} \right\}[/tex].
d) The proposition is [tex]\delta = \{\forall\,x,y \in \mathbb{N}\,|\,xP, P = \{x\,|\,x + y \le |x|+|y|\}\}[/tex].
The logical metalanguage involves the use of propositions, quantifiers, elements, predicates, logical connectives, mathematical operators and domains, which show clearly the mental scheme without any sign of subjectivity, typical in human language. Now we proceed to translate each sentence into logical metalanguage:
a) The product of two negative integers is positive.
[tex]\alpha = \left\{ \forall \, x, y \in \mathbb{N} \land \forall \,x, y < 0\,|\,\,xyP, P = \{(x,y)\,|\,x\cdot y > 0\}\right\}[/tex] (1)
b) The average of two positive integers is positive.
[tex]\beta = \left\{ \forall \, x, y \in \mathbb{N} \land \forall \,x, y > 0\,|\,\,xyP, P = \{(x,y)\,|\,x\cdot y > 0\}\right\}[/tex] (2)
c) The difference of two negative integers is not necessarily negative.
[tex]\gamma = \left\{\exists \, x, y \in \mathbb{N}\,\land\,\exists \,x,y < 0\,|\,xyP, P = \{(x,y)\,|\, x-y > 0\} \right\}[/tex] (3)
d) The absolute value of the sum of two integers does not exceed the sum of the absolute values of the integers.
[tex]\delta = \{\forall\,x,y \in \mathbb{N}\,|\,xP, P = \{x\,|\,x + y \le |x|+|y|\}\}[/tex] (4)
To learn more on propositions, we kindly invite to check this verified question: https://brainly.com/question/6709166
