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1 answer:
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Step-by-step explanation:
To show that a statement is a tautology using truth table - is to show that all the entries in the expression are truths T.
We can do this by taking each statement, expression by expression. For example, to show that
[~p ∧ (p ∨ q)] → q
is a tautology, knowing we have 3 columns, we have 2^3 = 8 rows. We start by putting putting truth values for p, q, and r respectively
Next, we find ~p, then find (p ∨ q), then find ~p ∧ (p ∨ q), before finally arriving at the required [~p ∧ (p ∨ q)] → q
TERMINOLOGIES AND SYMBOLS
- T means Truth
- F means False
- ~p means negation of p.
~p is F if p is T, and vice versa
- ∧ means conjunction.
p ∧ q is T only if p is T and q is T.
- ∨ means disjunction.
p ∨ q is T if either p or q is T.
- → is for conditional 'if then'
p → q is T if both p and q are T, or both p and q are F, or p is F and q is T, otherwise, it is F.
THE STEP BY STEP WORKINGS FOR THE STATEMENTS GIVEN ARE IN THE ATTACHMENT.
Answer:
The width and the length of the pool are 12 ft and 24 ft respectively.
Step-by-step explanation:
The length (L) of the rectangular swimming pool is twice its wide (W):
Also, the area of the walkway of 2 feet wide is 448:
Where 1 is for the swimming pool (lower rectangle) and 2 is for the walkway more the pool (bigger rectangle).
The total area is related to the pool area and the walkway area as follows:
(1)
The area of the pool is given by:
(2)
And the area of the walkway is:
(3)
Where the length of the bigger rectangle is related to the lower rectangle as follows:
(4)
By entering equations (4), (3), and (2) into equation (1) we have:
By solving the above quadratic equation we have:
W₁ = 12 ft
Hence, the width of the pool is 12 feet, and the length is:
Therefore, the width and the length of the pool are 12 ft and 24 ft respectively.
I hope it helps you!