Negate the following. (i) 4 is an even integer or 7 is a prime number

1.	Negate the following. (i) 4 is an even integer or 7 is a prime number. (ii) He likes to run and he does not like to sit. (iii) He likes Mathematics and he does not like Logic. (iv) If 6 is a divisor of 120 then 486 is not dividiable by 6. (v) If 2 triangles are similar then their ares are equal. (vi) It is cold or it is raining,
Answer» (i) Let p: 4 is even integer, q: 7 is a prime number given (p v q) ~(p ∨ q) = ~ p ∧ ~q 4 is not an even integer & 7 is not a prime number (ii) Let p: He likes to run, q: He likes to sit Given is ~ p ∧ ~q ∴ ~ (p ∧~ ) = ~p ∨ ~(~q) ≡ ~p ∨ q He does not like to run or he likes to sit (iii) Let p: He likes mathematices, q: He like logic Given (p ∧ ~q) ~ (p ∧ ~q) ≡ ~p ∨ q He does not like mathematics or he likes logic (iv) Let p: 6 is a divisor or 120 q: 486 is divisible by 6 Given (p → ~ q) ~(p → ~ q) = p ∧ ~(~q) (v) Let p: 2 triangles are similar. q: Areas are equal Given (p → q) ∴ ~(p → q) = (p ∧ ~q) 2 triangles are similar & areas are not equal. (vi) p: It is cold q: It is raining Given p ∨ a ~(p ∨ q) =~p∧~q It is not cold & it is not raining

Negate the following. (i) 4 is an even integer or 7 is a prime number. (ii) He likes to run and he does not like to sit. (iii) He likes Mathematics and he does not like Logic. (iv) If 6 is a divisor of 120 then 486 is not dividiable by 6. (v) If 2 triangles are similar then their ares are equal. (vi) It is cold or it is raining,

Answer»

(i) Let p: 4 is even integer, q: 7 is a prime number given (p v q) ~(p ∨ q) = ~ p ∧ ~q

4 is not an even integer & 7 is not a prime number

(ii) Let p: He likes to run, q: He likes to sit Given is ~ p ∧ ~q

∴ ~ (p ∧~ ) = ~p ∨ ~(~q) ≡ ~p ∨ q

He does not like to run or he likes to sit

(iii) Let p: He likes mathematices, q: He like logic

Given (p ∧ ~q) ~ (p ∧ ~q) ≡ ~p ∨ q

He does not like mathematics or he likes logic

(iv) Let p: 6 is a divisor or 120 q: 486 is divisible by 6

Given (p → ~ q) ~(p → ~ q) = p ∧ ~(~q)

(v) Let p: 2 triangles are similar. q: Areas are equal

Given (p → q)

∴ ~(p → q) = (p ∧ ~q)

2 triangles are similar & areas are not equal.

(vi) p: It is cold q: It is raining

Given p ∨ a ~(p ∨ q) =~p∧~q

It is not cold & it is not raining