Answer:
The two statements are logically equivalent.
Step-by-step explanation:
Let X be the statement that Bob is a double math and computer science major.
Y be the statement that Ann is a maths major.
Z be the statement that Ann is a double maths and computer science major.
The two statements written in terms of X, Y and Z now
a. Bob is a double math and computer science major and Ann is a math major, but Ann is not a double math and computer science major.
(x and y) and not z
b. It is not the case that both Bob and Ann are double math and computer science majors, but it is the case that Ann is a math major and Bob is a double math and computer science major.
not (x and z) and (x and y)
Noting that for logical statements,
Negation is represented by ~
And is represented by conjunction sign Λ
Or is represented by disjunction sign V
(x and y) and not z
(x Λ y) Λ (~z)
not (x and z) and (x and y)
~(x Λ z) Λ (x Λ y)
We can then simplify the second statement to obtain the first statement and prove the equivalence of both sides
~(x Λ z) Λ (x Λ y)
Using DE MORGAN'S theory, ~(x Λ z) = (~x) V (~z)
~(x Λ z) Λ (x Λ y) = ((~x) V (~z)) Λ (x Λ y)
Then applying the distributive law to the expression, we can open the bracket up
((~x) V (~z)) Λ (x Λ y)
= ((~x) Λ (x Λ y)) V ((~z) Λ (x Λ y))
Opening the first bracket up further
((~x) Λ (x Λ y)) V ((~z) Λ (x Λ y))
= ((~x) Λ x) Λ y) V ((~z) Λ (x Λ y))
The NEGATION law shows that (~x Λ x) = c (where c is a negation law parameter for when two opposite statements are combined in this manner, it works like a 0 in operation)
((~x) Λ x) Λ y) V ((~z) Λ (x Λ y))
= (c Λ y) V ((~z) Λ (x Λ y))
But (c Λ y) = (y Λ c) = c (according to the UNIVERSALLY BOUND law, see how c works like a 0 now?)
(c Λ y) V ((~z) Λ (x Λ y))
= c V ((~z) Λ (x Λ y))
= ((~z) Λ (x Λ y)) V c (commutative)
And one of the foremost IDENTITY laws is that (any statement) V c = c
((~z) Λ (x Λ y)) V c
= ((~z) Λ (x Λ y))
= (x Λ y) Λ (~z)
Which is the same as the first statement!
PROVED!!!
Hope this Helps!!!
