MATH121: Discrete Mathematics
Autumn 2004
Logic Problems
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Write the following statements in symbolic form using connectives. Make sure you dene appropriate
simple statements P; Q; R; : : :.
(i) Stocks are increasing but interest rates are steady.
(ii) Neither are stocks increasing nor are interest rates steady.
(iii) John is healthy and wealthy but not wise.
(iv) John is not wealthy but he is healthy and wise.
(v) John is neither healthy, wealthy nor wise.
(vi) John is healthy, wealthy and wise.
(vii) Either Olga will go out for tennis or she will go out for track but not both.
(viii) Freeze or I'll shoot.
(ix) I am on time for work if I catch the 8:05 bus.
(x) If it is after ve, then the meeting is over.
(xi) Either my watch is wrong or we are going to be late.
(xii) If Pat arrives early and Quinn stays late, then Robert is happy.
(xiii) If either Pat arrives early or Quinn does not stay late, then Robert is not happy.
(xiv) If Robert is not happy, then Pat does not arrive early.
(xv) Pat arrives early if and only if Quinn stays late, but it is not the case that if Quinn does not stay
late then Robert is happy.
Number each connective in order of importance in each of the following compound statements. State the
main connective
in each case.
(i) (P () Q) () ( R ^ S ) _ T
(ii) (R _ S ) ^ (P _ Q) =) (P =) Q)
(iii) (P =) Q) ^ (Q =) P ) () (P () Q)
(iv) (P =) Q) ^ (Q =) R) _ (R =) S )
(v)
P _ (R ^ S )
=) (Q _ T ) _ (P ^ S )
Let P , Q and R be statements. Construct truth tables for the following statement forms.
(i) P ^ Q
(ii) (P ^ Q) _ (P _ Q)
(iii) P ^ (Q ^ R)
(iv) P ^ ( Q _ R)
(v) P _ ( P _ Q) ^ (Q ^ R)
(vi) ( P _ Q) =) Q
(vii) (P _ Q) _ ( P ^ Q) =) Q
(viii) (P ^ Q) =) R
(ix) ( P _ Q) =) R
(x) (P ^ R) () (Q _ R)
(xi) (P =) R) () (Q =) R)
(xii) P =) (Q =) R) () (P ^ Q) =) R
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Let P , Q and R be statements. By nding truth tables, determine which of the following pairs of
statements are logically equivalent.
(i) P _ (P ^ Q) and P
(ii) (P ^ Q) and P ^ Q
(iii) (P ^ Q) _ R and P ^ (Q _ R)
(iv) ( P _ Q) ^ (P _ R) ^ ( P _ Q) and (P _ R)
(v) R _ P ) ^ ( R _ (P ^ Q) ^ (R _ Q) and P ^ Q
(vi) P =) (Q _ R) and (P ^ Q) =) R
(vii) P =) (Q =) R) and (P =) Q) =) R
Assume x is a real number. Use DeMorgan's laws to write negations for the following statements.
(i) 2 < x < 7
(ii) 10 < x < 2
(iii) 1 > x 3
(iv) 0 > x 7
Using Theorem 1.4.2, part 6, write negations for each of the following statements.
(i) If S is a square, then S is a triangle.
(ii) If today is New Year's Even, then tomorrow is January.
(iii) If the decimal expansion of x is terminating, then x is rational.
(iv) If n is prime, then n is odd or n is 2.
(v) If Tom is Ann's father, then Jim is her uncle and Sue is her aunt.
Let P , Q and R be statements. Determine whether the following statements are tautologies, contradictions
or contingent statements.
(i) (P ^ Q) _ P _ (P ^ Q)
(ii) (P ^ Q) ^ ( P _ Q)
(iii) ( P _ Q) ^ (Q ^ R) ^ Q
(iv) ( P _ Q) _ (P ^ Q)
Using Theorem 1.4.2 and the conclusions from Tutorial Sheet Week 2, prove the following logical equivalences.
(i) (P ^ Q) _ (P ^ Q) P
(ii) (P _ Q) ^ ( P _ Q) Q
(iii) (P _ Q) _ ( P ^ Q) P
Using Theorem 1.4.2, rewrite the following statements using only the connectives ^ and .
(i) (P ^ Q) =) R
(ii) (P _ Q) =) (R _ Q)
(iii) (P =) R) () (Q =) R)
(iv) P =) (Q =) R) () (P ^ Q) =) R
Use this exercise to practice the \quick method"; avoid using truth tables if you can.
Determine which of the following statements are tautologies.
(i) (P =) Q) =) (Q =) P )
(ii) (P ^ Q) =) (P () Q)
(iii) (P ^ Q) =) R _ ( P _ Q)
(iv) P =) ( Q ^ R) _ (P ^ R)
(v) (P ^ Q) =) R =) ( R =) P )
(vi) (P ^ Q) _ R =) (Q _ R)