Convert the following statements into the formal notation of propositional logic
ID: 1720414 • Letter: C
Question
Convert the following statements into the formal notation of propositional logic (i.e. using variables and logical operators). Make sure to explain what each variable you introduce represents. Whenever we add a rational number and an irrational number, the sum is irrational. Two integers are odd only if their sum is even. It is necessary that a|(b + c) be true for a|b and a|c to be true. For (ac)|(bd) to be true , it is sufficient that a|b and c|d. For x to be an odd number, it is necessary and sufficient that x - 1 is even. An integer is even if and only if its square is even.Explanation / Answer
Answer :
(1)Consider the given statement "when ever we add a rational number and an irrational number , the sum is irrational "
Let x be a rational number and y be an irrational number Let us denote
P(x,y) : add x and y
Q (x,y) : x+y is irrational
The the symbolic form of the given statement is : (x) (y) P(x,y) Q(x,y)
(2) Consider the given statement " Two integers are odd only if their sum is even "
Let us denote
P : The sum of two integers is even
Q : Two integers are odd
The the symbolic form of the given statement is : P Q
(3) Consider the given statement " It is necessary that a|(b+c) be true for a|b and a|c to be true "
Let us denote
P : a| b
Q : a|c
R : a|(b + c)
The the symbolic form of the given statement is : (P Q) R
(4) Consider the given statement " For (ac)| (bd) to be true ,if is sufficient that a|b and c|d "
Let us denote
P : (ac)| (bd)
Q : a|b
R : c|d
The the symbolic form of the given statement is P (Q R)
(5) Consider the given statement " for x to be an odd number it is necessary and sufficient that x - 1 is even "
Let us denote
P(x) : x is odd
Q(x) : x - 1 is even
The the symbolic form of the given statement is ( x ) P(x) Q(x)
(6) Consider the given statement " An integer is even if and only if its square is even "
Let us denote
P(x) : x is even
Q(x) : x2 is even
The the symbolic form of the given statement is ( x ) P(x) Q(x)