1) Structural Induction, Induction
2) Logically Valid of
![[;\forall;]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_sWvj5QBL_b5HjCYIKaNze7D3tMvUi9fsPuqavZMZfH91fxeTsiyNkiq8BClbjoCXOiSR2Jovmm5SkD_Dp4YZYPeA=s0-d "\forall")
and ![[;\exists;]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vBW57wg4HBCrqkPn8iE8PZezU4rVQkZNEcSzh63yyG-gn06hWdzxrrKgWlGzNSfhta3_oD92W6gZ0yWodb6dAXig=s0-d "\exists")
in ![[; \exists z R(x,z) and \forall y R(x,f(x,y);]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_ukXLWt2VSXh8OHMI7Vz0D8JhjFqm_AcHVyDnLWCnnEZxmm3TO071cefWTQt4aAeGyQfkNnN3j4In5Luea9s9Lt7HKGuhLFoh-OqTXOocMezh-14hetyuP_b6ud6wkaVcCX52mGb2ObJD0P2UfnqT6LFX-r0v3pNEgfz8DoXnA=s0-d "\exists z R(x,z) and \forall y R(x,f(x,y)")
3)
![[; L \cap K;]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vWCeN8Qo2ZjZtTmqjXXNY47tg9YIoglkEbyUdzD9QERDd1G3rbha9ZxoOEawouLZRTLeosrMB8gqKKS2YHZpmTYcWH3uEAFQQ=s0-d "L \cap K")
, ![[;L \cup K;]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_sB6AVBA5SIpFLefiBSixGOwHBQ10hRowYaYV3QukRJsRfEsT0XIiZ0gPmp7bg23rypEpK6VXXYIQ8d3s7rSRqhbOQgoWEY=s0-d "L \cup K")
![[; L -K ;]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_uMaY4Ap4y5MS36ah22USqj1okTvoENkkyBShyBjigeYBXDyKxEd60sa-o6rL794JU-ojrssBP96DCb7eR7V_BGIg=s0-d "L -K")
: context free closure4)
![[; f: {a^nb^{f(n)} | n \in N;]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vF-w2Dch9vx31hSLFl1dTbu7Xda8CHPTqRgJDVYgIaRL6PhvkAAwDqQRc1mMIT0xR2ZAMVD8SfnkqCDoq8PGm8C6m6kYa5LZj4uwDzDuLvygRGVMGy9umdafEdQ0qzONlOo3YFEJMAEzmgNzhtS60=s0-d "f: {a^nb^{f(n)} | n \in N")
is not regularand
![[;f: a^nb^{kn}| n,k \in N;]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_ukGr89YJ-iWn8V3z6HcgvxKYwmNDWin_gDoSZujl0I1FWf_o4bENbW8uD1R6ajbu6cgGqKuwzVLtFFQ8v5OWN7RTSQgzO5FaeYr8XBAMl1znwZSiVE43Zz25VLqV9hQWvQGA=s0-d "f: a^nb^{kn}| n,k \in N")
is context free5) The function f(n) :the number of perfect square numbers that < n,
f(n) is Primitive Recursive
6) The set of {m |p+q, when
![[;p \in P;]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vAMhswP1kgJv76wVKshK_MAkDVWh2c8w3UMBkf2vt3ezigxAqAlgqdhoil1-JREnSPgrI41iVkbY1FZkDBlAdU4wn94Q=s0-d "p \in P")
and ![[; q \in Q;]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vgRDffnlIhypwj7ard5ndXYfN5AUjON2uvTQ-zAzDWDppoumDproHh8Vt8-H6cQhTqEd-dcQmEgPthxeXabMiIt-Tq1ZbqAqY=s0-d "q \in Q")
and P and Q are recursive enumerable sets } is a recursive enumerable set.
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