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Home > College Algebra calculators > Mathematical Logic, truth tables, logical equivalence example
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Mathematical Logic, truth tables, logical equivalence example
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1. Examples
The implication that statement `S` follows from statements `S_1,S_2,S_3,...,S_n` is called an argument.
Thus the statements `S_1^^S_2^^S_3^^...^^S_n=>S` is an argument.
In symbols argument is represented by `(S_1,S_2,S_3,...,S_n;S)`. In this argument `S_1,S_2,S_3,...,S_n` is called hypothesis and S is called conclusion.
Such an argument is said to be logically valid if the conclusion `S` is true whenever the hypothesis statements `S_1,S_2,S_3,...,S_n` are all true. Not all arguments are logical.
To test the logical validity of argument `(S_1,S_2,S_3,...,S_n;S)`, we must form a truth table.
The rows in which all statements `S_1,S_2,S_3,...,S_n` have truth value T int the truth table are called critical rows.
If the conclusion `S` has truth value T in every critical rows of the truth table of an argument then we say that the argument is logically valid.
1. Examine the logical validity of the following argument
Hypothesis : p or q; not p and Conclusion : q
Solution:
Hypothesis :
`S_1:pvvq`
`S_2:~p`
Conclusion :
`S : q`
| `(1)` | `(2)` | `(3)=(1)vv(2)` | `(4)=~(1)` | | `(5)` | | `p` | `q` | `S_1`
`pvvq` | `S_2`
`~p` | | `S`
`q` | | T | T | T | F | | T | | T | F | T | F | | F | | F | T | T | T | critical row | T | | F | F | F | T | | F |
The conclusion `(S)` is true in all critical rows. So the argument is logically valid.
2. Examine the logical validity of the following argument
Hypothesis : (p and not(q)) => r;p or q;q => p and Conclusion : r
Solution:
Hypothesis :
`S_1:(p^^~q)=>r`
`S_2:pvvq`
`S_3:q=>p`
Conclusion :
`S : r`
| `(1)` | `(2)` | `(3)` | `(4)=~(2)` | `(5)=(1)^^(4)` | `(6)=(5)=>(3)` | `(7)=(1)vv(2)` | `(8)=(2)=>(1)` | | `(9)` | | `p` | `q` | `r` | `~q` | `p^^~q` | `S_1`
`(p^^~q)=>r` | `S_2`
`pvvq` | `S_3`
`q=>p` | | `S`
`r` | | T | T | T | F | F | T | T | T | critical row | T | | T | T | F | F | F | T | T | T | critical row | F | | T | F | T | T | T | T | T | T | critical row | T | | T | F | F | T | T | F | T | T | | F | | F | T | T | F | F | T | T | F | | T | | F | T | F | F | F | T | T | F | | F | | F | F | T | T | F | T | F | T | | T | | F | F | F | T | F | T | F | T | | F |
The conclusion `(S)` is not true in all critical rows. So the argument is not logically valid.
3. Examine the logical validity of the following argument
Hypothesis : p => q;q => r and Conclusion : p => r
Solution:
Hypothesis :
`S_1:p=>q`
`S_2:q=>r`
Conclusion :
`S : p=>r`
| `(1)` | `(2)` | `(3)` | `(4)=(1)=>(2)` | `(5)=(2)=>(3)` | | `(6)=(1)=>(3)` | | `p` | `q` | `r` | `S_1`
`p=>q` | `S_2`
`q=>r` | | `S`
`p=>r` | | T | T | T | T | T | critical row | T | | T | T | F | T | F | | F | | T | F | T | F | T | | T | | T | F | F | F | T | | F | | F | T | T | T | T | critical row | T | | F | T | F | T | F | | T | | F | F | T | T | T | critical row | T | | F | F | F | T | T | critical row | T |
The conclusion `(S)` is true in all critical rows. So the argument is logically valid.
4. Examine the logical validity of the following argument
Hypothesis : p => q;p => r and Conclusion : p => (q and r)
Solution:
Hypothesis :
`S_1:p=>q`
`S_2:p=>r`
Conclusion :
`S : p=>(q^^r)`
| `(1)` | `(2)` | `(3)` | `(4)=(1)=>(2)` | `(5)=(1)=>(3)` | `(6)=(2)^^(3)` | | `(7)=(1)=>(6)` | | `p` | `q` | `r` | `S_1`
`p=>q` | `S_2`
`p=>r` | `q^^r` | | `S`
`p=>(q^^r)` | | T | T | T | T | T | T | critical row | T | | T | T | F | T | F | F | | F | | T | F | T | F | T | F | | F | | T | F | F | F | F | F | | F | | F | T | T | T | T | T | critical row | T | | F | T | F | T | T | F | critical row | T | | F | F | T | T | T | F | critical row | T | | F | F | F | T | T | F | critical row | T |
The conclusion `(S)` is true in all critical rows. So the argument is logically valid.
This material is intended as a summary. Use your textbook for detail explanation.
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