Question

Show that the following are tautologies by going through a series of equivalent propositions until you reach T.

(a) [(p → q) ∧ (q → r)] → (p → r)

(b) [(p ∨ q) ∧ (p → r) ∧ (q → r)] → r

please use laws

Thank You

Answer 1

(1) By applying logical equivalence twice, [(p>0)^(9>»)]=(p >>) = [6-pva) A(- www] → (-p v~) -[laboval AG 9Vx)] vlab V) Using De Morgan's law twice and double negation law'. Salapvalvil-qur)] V(-pvr) = [(pica) v (94-)] V-18 Vr) Using distributive land twice, = [(pVg140)1 67 9 Vign-v)] v(ab vy) = [(pvq) (p Varl) All Aquq) (19V+r)]]v (Gp Vr) Using negation law and commutative laws = Opcep va) A(PV-v))^(19Nan] V (15 Vv) Using identity low = [PNG) A (PVar))^(-iqv+r)] v(up Vr) Using commntative law and associative law: = [((bova) A(- 90 - Y))^(V-)] vepvr) Using distributive law = [(($va)^(-91-r))/(-pur)] A[Cpvar) v (-þ Vx)] Using commutative and associative la = [((p Va) ^ ( 9 V-v)) v (- p vv)]^ [CÞV-p) V (VA) Using negation law, = 'ccpvaj) 169 Var)) (-þv»)](T Vir V-v)] Using commutative law and domination low, = [(apva) A(-gv-r)) v (npvr] AT Using identity law, = ((pva) (og V-r)) V(+ pvm) Using distributive law, = ((pva) v(-þvv)]^(19^ *v) v (+b Vr)) Using commutative and associative lano, =((pv-p) (qv)) ((-qa-pv (arvr) Using negation love and associative law, =lp vap) v (qvol) Allaq v-op) Vrs)) Using negation law and commutative law, I = Gvq vr)) a (1xq Vap) VT)

STAT Using distributive law, Using domination law, Using identity law (with p=T), OTO The conditional statement is equivalent with time T, this the conditional statement is a tautology (6) By using logical equivalence, cf vaja?p=09 169+r)]*x = [(pvajal-pur)^(1 9V m)] -→Y = [(b Vq) A (C-160-g) Vr)] → r. Using logical equivalence, - (PV gy) (-pang) vy)] Vy Using De Morgan's land thrice) and on third time also wsing double negation law, = [-pwa) Vall-pang) Vr)]VY = [(apang)(-(-pang) A-r)] Vr = [(-panav (ep vg) Ar)] Vr Using distributive law, 8 = [(l-ph-q) v (PV q )) ((-pa-q) W-7)] Vr Using commutative and associative law, 8 = [ll-papu(igua)) (( PA-9) Var]] Nr Using negation law, = (TV(-quq)) A (lapa-q) +r)] Vr. Using domination law, =[TAC-pa-q]N-r)] vr Using identity and, I = [(ap 1-q) var] Vr Using distrubutive law, =(op1-9) (rvr) Using negation and commutative law, 0 0= (b1-91) VT Vring domination law, zT The conditional statement is equivalent with teme T, thus the conditional statement is a tautology.