1. Write negations for the following statements. Do not use the expressions \"it
ID: 3124115 • Letter: 1
Question
1. Write negations for the following statements. Do not use the expressions "it is not the case that" or it is not true that" in your final answer (a) The sun is shining and the birds are singing (b) If n is an integer, then n+1 is not an integer (c) Every integer is divisible by a prime. 2. Write the converse and the contrapositive of the following implications (a) If a number is divisible by 12, then it is divisible by 3. b) The sum of two odd numbers is even 3. Suppose we have statements PQ, R such that (PAQ) R is false. Determine, with some explanation, the truth values of (a) P (b) (NQ) R (c) R P (d) RA (PVQ) 4. You know that for a certain choice of truth values, the expression Q V (NP) (PAR) V S1 is false. Without using a truth table, determine the truth values of P, Q, R, and S. Clearly explain all reasoning. 5. (a) Use a truth table to show that P (QVR) PA (n Q) R. (b) Show that P (QVR) PA (N Q) R using properties of logical equivalences.Explanation / Answer
(According to Chegg policy, only four questions will be answered. Please post the remaining in a separate question)
1. (a) p: The sun is shining q: The birds are singing
~(p^q) = ~p V ~q
=> Negation is: The sun is not shining OR the birds are not singing.
(b) If n is an integer, then n/(n+1) is not an integer.
The negation is: If n is an integer, then n(n+1) is an integer.
(c) Every integer is divisible by prime
p: Divisible by prime
q: x Z, p
=> ~q: x Z, ~p
The negation is: Some integers are not divisible by p.
2. (a) If a number is divisible by 12, then it is divisible by 3.
p: Divisible by 12 q: Divisible by 3.
Given statement is p->q
Converse is q -> p i.e If a number is divisible by 3, it is divisible by 12.
Contrapositive is ~q -> ~p i.e If a number is not divisible by 3, it is not divisible by 12.
(b) The sum of two odd numbers is even.
p: Numbers are odd q: Sum is even
Given statement is p->q
Converse is q -> p i.e If the sum of two numbers is even, they are odd.
Contrapositive is ~q -> ~p i.e If the sum of two numbers is not even, they are not odd.
3. (P^Q) -> R is false.
The only way a -> b is false is if a is true and b is false.
=> (P^Q) is true and R is false.
Also the only way if a^b is true is if both a and b are true.
=> P is true, Q is true and R is false.
(a) P -> Q : True -> True = True
(b) (~Q)VR: False V False = False
(c) R -> P: False -> True = True
(d) R ^ (PVQ): False ^ (True V True) = False
4. [ Q V (~P) ] -> [ ~(P ^ R) V S ] is false.
a -> b is false only if a is true and b is false.
=> Q V (~P) is true (1)
~(P ^ R) V S is false. (2)
Observe (2)
a V b is false only if both a and b are false
=> ~(P ^ R) is false and S is false
=> P ^ R is true and S is false.
a ^ b is true only when both a and b are true
=> P is true, R is true, S is false.
From (1)
=> Q V (~P) is true.
Since P is true, ~P is false
=> Q V false = true
=> Q is true.
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